Form and fabrication of semiconductor-superconductor nanowires and quantum devices based thereon

ABSTRACT

The disclosure relates to a quantum device and method of fabricating the same. The device comprises one or more semiconductor-superconductor nanowires, each comprising a length of semiconductor material and a coating of superconductor material coated on the semiconductor material. The nanowires may be formed over a substrate. In a first aspect at least some of the nanowires are full-shell nanowires with superconductor material being coated around a full perimeter of the semiconductor material along some or all of the length of the wire, wherein the device is operable to induce at least one Majorana zero mode, MZM, in one or more active ones of the full-shell nanowires. In a second aspect at least some of the nanowires are arranged vertically relative to the plane of the substrate in the finished device.

CROSS REFERENCE TO RELATED APPLICATION

This patent application claims the benefit of U.S. Provisional PatentApplication No. 62/701,458, filed Jul. 20, 2018 and entitled “Form AndFabrication Of Semiconductor-Superconductor Nanowires And QuantumDevices Based Thereon”, the disclosure of which is hereby incorporatedby reference.

BACKGROUND

Quantum computing is a class of computing in which inherently quantummechanical phenomena, such as quantum state superposition andentanglement, are harnessed to perform certain computations far morequickly than any classical computer could ever be capable of. In a“topological” quantum computer, calculations are performed bymanipulating quasiparticles—called “non-abelian anyons”—that occur incertain physical systems. Anyons have unique physical characteristicsthat distinguish them from both fermions and bosons. Non-abelian anyonsalso have unique properties with respect to abelian anyons. It is theseunique properties that serve as a basis for topological quantumcomputing, in which information is encoded as a topological property ofnon-abelian anyons; specifically the braiding of their space-timeworldlines. This has certain benefits over other models of quantumcomputation. One key benefit is stability, as the quantum braiding isunaffected by perturbations on a scale that could cause error-inducingquantum decoherence in other types of quantum computer.

A number of types of physical system have been considered as potentialhosts of non-abelian anyons, such as “5/2 fractional quantum Hall”systems in condensed matter physics, and systems of topologicalinsulators in contact with superconductors. Another example issemiconductor-superconductor (SE/SU) heterostructures such as SE/SUnanowires. With regard to these, a key advance in the field was therealization that non-abelian anyons, in the form of “Majorana zeromodes” (MZMs), can be formed in regions where semiconductor (SE) iscoupled to a superconductor (SU). Based on this phenomenon, a smallnetwork of SE/SU nanowires can be used to create a quantum bit, whereineach SE/SU nanowire comprises a length of semiconductor coated with asuperconductor.

A quantum bit, or qubit, is an element upon which a measurement with twopossible outcomes can be performed, but which at any given time (whennot being measured) can in fact be in a quantum superposition of the twostates corresponding to the different outcomes.

A “topological” qubit is a qubit implemented based on theabove-mentioned technology of non-abelian anyons in the form of MZMs. Anon-abelian anyon is a type of quasiparticle, meaning not a particle perse, but an excitation in an electron liquid that behaves at leastpartially like a particle. Particularly an anyon is a quasiparticleoccurring in a two-dimensional system (two degrees of freedom in space).A Majorana zero mode is a particular bound state of such quasiparticles.Under certain conditions, these states can be formed close to thesemiconductor/superconductor interface in an SE/SU nanowire network, ina manner that enables them to be manipulated as quantum bits for thepurpose of quantum computing. Regions or “segments” of the nanowirenetwork between the MZMs are said to be in the “topological” regime.

A Majorana-based qubit conventionally involves gating in order toexhibit such topological behaviour. That is, an electrical potential isapplied to a segment of the semiconductor of one of the nanowiresforming the qubit. The potential is applied via a gate terminal placedadjacent to the nanowire in the fabricated structure on the wafer. Amagnetic field is also required to induce the topological regime. Themagnetic field is applied from an electromagnet placed outside thewafer, typically within the refrigerating compartment as used to inducethe superconductivity in the superconductor.

Conventionally, building Majorana-based topological quantum computingdevices involves the formation of superconducting islands on thesemiconductor. Some parts of the superconductor are topological (T) andsome parts of which are non-topological (e.g., conventional S-wave (S)).The topological segment supports Majorana zero modes appearing at itsopposite ends. The existing techniques for realizing MZMs require strongmagnetic fields as well as electrostatic gating in order to drive thehalf-shell nanowires into the topological phase. The MZMs are induced bya coupling of the magnetic field to the spin component of the electrons.This requires a strong magnetic field.

In some fabrication techniques, the semiconductor of the nanowires maybe formed in the plane of the wafer by a technique such as selectivearea growth (SAG). The superconducting material may then be depositedselectively over the semiconductor, or may be deposited as a uniformcoating and regions subsequently etched away to form the islands.

Another method of fabricating a device comprisingsemiconductor-superconductor nanowires is disclosed in “Epitaxy ofsemiconductor-superconductor nanowires”, P. Krogstrup et al, NatureMaterials, 12 Jan. 2015, pages 400-406. The semiconductor cores of thenanowires are grown vertically relative to the plane of the wafer, andthen angle deposition is performed in order to deposit a coating ofsuperconductor on facets of the semiconductor core. The nanowires arethen “felled” by sonication and aligned in the horizontal plane by meansof optical microscopy. Parts of the superconductor coatings are thenetched away from the nanowires so as, in the resultant device, to leaveeach nanowire coated with just the superconducting islands mentionedabove.

SUMMARY

According to a first aspect disclosed herein, it is disclosed to coatthe full perimeter of each nanowire with superconductor, i.e. to providea full superconducting shell, and to leave this shell in place the finaldevice (rather than etching away portions of the superconductor ordepositing it selectively to form only superconducting islands over thesemiconductor of the wire). This full-shell coating may be implementedfor example by rotating the substrate or the deposition beam duringdeposition of the superconductor. With a full shell, the inventors havediscovered that only a relatively weak orbital magnetic field is neededto drive the system into the topological regime. The disclosed formationmay also be used to mitigate other technological challenges present inearlier schemes, in that it avoids electrostatic tuning into topologicalphase, protects topological elements from spatial inhomogeneities, andincreases charging energy of the superconducting island forming part ofthe qubit.

The disclosed formations can be used to build quantum computingstructures, such as to form topological qubits, qubit systems, andquantum computers based on full shell nanowires, including but notlimited to the example designs illustrated later. Example designsinclude “horizontal” and “vertical” designs.

According to a second aspect, there is disclosed a structure in whichvertically grown nanowires are left in the vertical orientation relativeto the wafer in the final device, rather than being felled and orientedhorizontally in the plane of the wafer as in the prior techniques. Anadvantage of this is that it allows for 3D integration. Currentapproaches require that the nanowires are laid out flat in a 2D networkin the plane of the wafer, but this takes up a lot of space. Insteadtherefore, the present disclosure discloses an approach in which thenanowires are left standing vertically in the orientation they weregrown in, and then the rest of the device is built around them.

The first and second aspects may be used together or independently. I.e.in the first aspect the nanowires may be left vertical or felled andaligned horizontally in the plane of the wafer; whilst in the secondaspect, the nanowires may be left as full-shell nanowires or may beetched to remove portions of the superconductor from the wires.

In either aspect the disclosed structure may be used to form atopological qubit or a topological computer or universal quantumcomputer comprising multiple qubits. In the second aspect, the disclosedstructure may even be useful in forming other, non-topologic quantumdevices.

Note that a “device” as referred to herein means a finished device, i.e.a device formed in a finished die or chip. Typically the device wouldalso be packaged, i.e. in an integrated circuit package.

Note also that terms such as “horizontal”, “vertical”, “bottom” and“top” as referred to herein are meant relative to the plane of the waferor substrate. I.e. horizontal means parallel to the plane of thesubstrate and vertical means perpendicular to the pane of the substrate,while bottom means at the end of the nanowire closest to the substrateand top means the end farthest from the substrate. These terms do notnecessarily imply anything about the orientation with respect togravity.

This Summary is provided to introduce a selection of concepts in asimplified form that are further described below in the DetailedDescription. This Summary is not intended to identify key features oressential features of the claimed subject matter, nor is it intended tobe used to limit the scope of the claimed subject matter. Nor is theclaimed subject matter limited to implementations that solve any or allof the disadvantages noted herein.

BRIEF DESCRIPTION OF THE DRAWINGS

To assist understanding of embodiments of the disclosure and to show howsuch embodiments may be put into effect, reference is made, by way ofexample only, to the accompanying drawings in which:

FIG. 1 is an illustration of a quantum system,

FIG. 1A is an illustration of a semiconducting nanowire with a fullsuperconducting shell,

FIG. 1B is an illustration of a semiconducting nanowire with a thinshell,

FIG. 2 is a topological phase diagram in a thin shell limit,

FIG. 3(a) is a topological phase diagram for a full core model,

FIG. 3(b) shows band structures at points indicated in FIG. 3(a),

FIG. 4(a) shows a Majorana coherence length for the full core model,

FIG. 4(b) shows a Majorana wavefunction integrated over radius,

FIG. 5(a) shows an example design for a tetron qubit (also referred toas quad qubit),

FIG. 5(b) shows an example design for a hexon qubit,

FIG. 6 shows an example design for a network of hexon qubits,

FIG. 7 shows a detail of FIG. 6,

FIG. 8 shows another example design for a network of hexon qubits,

FIG. 9 shows a detail of FIG. 8,

FIG. 10 shows an example design for a network of tetron qubits,

FIG. 11 shows another example design for a network of tetron qubits,

FIG. 12 shows an example fabrication process for fabricating verticalnanowires,

FIG. 13 shows an example design for bottom layer control of a verticalnanowire device,

FIG. 14(a) shows a side view of an example design having horizontalnanowires,

FIG. 14(b) shows a side view of another example design having verticalnanowires,

FIG. 14(c) shows a side view of another example design having verticalnanowires,

FIG. 15 is a sketch of the energy gap in a superconducting shell as afunction of B,

FIG. 16 shows a topological phase diagram of a hollow cylinder model

FIG. 17 shows a topological phase diagram for a full cylinder model,

FIG. 18 shows the evolution of local density states at the end of afinite wire,

FIG. 19A shows a topological phase diagram for a disordered fullcylinder model,

FIG. 19B shows effects of angular-symmetry-breaking perturbations,

FIG. 20 shows a simulation of a superconducting shell,

FIG. 21 shows a phase diagram of a hollow cylinder model,

FIG. 22 shows a probability density for lowest energy spin-up andspin-down modes,

FIG. 23 shows local density states in the middle of a wire,

FIG. 24A shows phase diagrams for different disorder realizations, and

FIG. 24B shows further phase diagrams for different disorderrealizations.

GLOSSARY OF MATHEMATICAL SYMBOLS

The following symbols are referred to herein:

-   {circumflex over (z)}: direction of the nanowire-   {right arrow over (r)}: direction radial to the nanowire-   R₁: (optional) insulating core radius-   R₂: semiconductor radius-   R₃: outer radius of superconducting shell-   R: thin shell radius-   {right arrow over (B)}: magnetic field applied to nanowire-   {right arrow over (A)}: electromagnetic vector potential-   H₀: Hamiltonian for the semiconducting core-   e: electric charge of an electron-   m*: effective mass-   μ: chemical potential-   α_(r) or α: strength of the spin-orbit coupling-   σ_(i): spin ½ Pauli matrices-   Δ₀: s-wave superconducting pairing potential-   Ψ: Nambu basis-   H_(BdG): Hamiltonian for the proximitized nanowire-   {right arrow over (p)}: momentum-   d: thickness of the superconductor shell-   λ_(L): London penetration depth-   ψ: magnetic flux-   Φ₀: magnetic flux quantum-   φ or ϕ: Angular coordinate-   n∈    : winding number-   n_(eff): prefactor of the effective Zeeman term-   τ_(i): Pauli matrices representing particle/hole degrees of freedom    in Nambu space-   m_(J) or m_(J): angular quantum number-   U: Unitary transformation-   E_(gap): bulk energy gap-   V_(Z): Zeeman energy-   Ec: charging energy-   L: angular momentum operator-   J_(z): orbital angular momentum-   Ψ: Majorana wavefunction-   ξ: frequency of coherence lengths-   m_(e): mass of an electron

DETAILED DESCRIPTION OF EMBODIMENTS

The present disclosure introduces calculations and simulations showingthat over a wide range of values of B field, spin-orbit coupling,chemical potential, radius and temperature, then semiconductor wireswith a full shell of superconductor can enter the topological phasewhich supports MZMs, the key element in the topological approach toquantum computation.

This prompted a flurry of further theoretical activity by the inventorscumulating in solid analytical verification of the topological phase,extensive numerical simulation of the parameters which determine accessto the phase, and practical characteristics of the phase (coherencelength and gap). Then with the full shell road to MZMs in hand, theinventors turned to the question, “Could this new full shell technologybe exploited in the design of qubits, arrays of qubits, and all the wayto a scalable architecture for a topological quantum computer?” Theinventors found that, indeed, the answer to all these questions is“yes”. This disclosure includes the theoretical work supporting thediscovery of scalable designs for topological quantum computer based onfull shell wires.

In fact there are at least two aspects to the present disclosure: I)what can be done with the now-known topological phase in full shellwires, and II) what can be done with vertical “forests” of vapor liquidsolid (VLS) grown nanowires as quantum computing elements.

A linkage between I) and II) is that, in developing a scalabletechnology for full shell wires, the inventors realized that since thewires grow (VLS) in beautiful, orderly, vertical arrays, it would be apity to ever have to knock them over. It is therefore disclosed to leavethe forest in place and build the rest of the topological quantumcomputer around it. (Though horizontal designs are also possible in thefirst aspect.) Once headed down the path of achieving 3D integration ofquantum computing elements though vertical forests of VLS wires, theinventors realized that there were fruitful applications even outsidethe topological realm. Conventional condensed matter qubits, such astransmons and gatemons will have limited fidelity and will have to beproduced in the millions for quantum computers based on these to be ableto support the necessary layers of error correction. It is recognizedherein that vertical forests of semiconductor wires have the potentialto be used to form dense arrays of majorana based qubits or conventionalqubits.

FIG. 1 gives a schematic illustration of an example system 1 inaccordance with embodiments of the present disclosure. For example thesystem 1 may be a quantum computer such as a topological quantumcomputer or universal quantum computer. The system 1 comprises a device2 in the form of a die (chip) disposed within a refrigerated chamber 3.The die may be packaged in an integrated circuit (IC) package (notshown) and may be connected to external equipment (also not shown) viapins of the package, the equipment being arranged to control and/or takemeasurements from the device via contacts between the interior andexterior of the device 2. The system 1 further comprises anelectromagnet 4, typically disposed within the refrigerated chamber 3.

The device 2 has been formed from a wafer using layered fabricationprocesses. The device 2 comprises a substrate 5 which defines a plane.The device 2 further comprises one or more layers 6 formed over thesubstrate 5. These may comprise for example a dielectric layer formedover the substrate 5, a layer of semiconductor material formed over thedielectric layer, a layer of semiconductor material formed over thesemiconductor layer, and a layer of filler material formed over thesuperconductor layer.

The device 2 comprises one or more semiconductor-superconductornanowires 7, preferably a plurality of such wires. The wires 7 areembedded in one or more of the layers 6. Each of the nanowires 7comprises a length of semiconductor material 9 (the core of the nanowire7) defining an axis. The core may be circular, elliptical or polygonalin cross section (i.e. the shape in the lane perpendicular to the axisof the wire). Each nanowire 7 further comprises a coating ofsuperconducting material 10 formed over at least part of the exterior ofthe length of the nanowire. According to the first aspect disclosedherein, the superconductor material 10 is formed all the way around theperimeter (e.g. circumference) of the semiconductor core 9 in the planeof the axis (i.e. in cross section), along part or all of the length ofthe nanowire (part of all of the way along the axis in the directionparallel to the axis). This is what is meant by a full-shell nanowire.One, some or all of the nanowires 7 in the device 2 may take the form offull-shell nanowires.

To fabricate such formations, the semiconductor cores 9 of the nanowires7 may be grown vertically for example using the growth method previouslydisclosed by Krogstrup et al, as cited in the background section.Vertical here means perpendicular to the plane of the substrate 5. Thesuperconductor material 10 may then be deposited on the core 9 by meansof an angled beam deposition technique. To form the full shell coating,either the wafer or the beam may be rotated during the deposition of thesuperconductor 10. The semiconductor material 9 is preferably a materialwith a high spin-orbit coupling, such as InAs (indium arsenide) or InSb(Indium antimonide). The superconductor material 10 is preferably ans-wave superconductor such as Al (aluminium) or Nb (Niobium).

According to the second aspect disclosed herein, the vertical nanowires7 may be left in the vertical position in the finished device 2. Tophysically support the vertical nanowires, the deposited layers 6include one or more layers of filler material, which may for example bea plastic or wax. E.g. the filler may be the plastic PMMA.

Alternatively some or all of the nanowires 7 (e.g. full-shell nanowires)may be felled and have their axis aligned in the horizontal plane(parallel to the substrate 5) in accordance with the previously knownapproach disclosed by Krogstrup et al.

In the final system 1, then one, some or all of the nanowires 7 a (e.g.some or all of the vertical nanowires) are used as active nanowires,i.e. operative quantum components of a quantum structure such as aqubit. For example one, some or all of the nanowires 7 may be used toinduce Majorana zero modes within the nanowire 7, which can be used toform a structure that acts as a Majorana based qubit. As will bediscussed in more detail later, in embodiments bunches of verticalnanowires 7 a can be used to form qubit structures such as quad qubits(tetron qubits) or hexon qubits. In other embodiments the nanowires 7could be used to form other kinds of qubit such as transmon or gatemonbased qubits.

In embodiments all of the nanowires 7, or at least all of the activenanowires 7 a, may be aligned with their axes in the same direction,i.e. parallel to one another.

Formed in one or more of the layers 6 are one or more layers ofcircuitry 8, which may for example comprise one or more gates,interconnects, semiconductor networks, and/or other electrical contacts8. In embodiments, some or all of any semiconductor used in thecircuitry 8 may be formed from the same semiconductor material 9 as thecores of the nanowires 7. In embodiments, some or all of any conductorsused in the circuitry 8 may be formed from the same superconductormaterial 10 as the coating of the nanowires 7. The layer(s) of circuitry8 may for example be arranged to enable any or all of: connectingtogether the nanowires 7 into quantum structures such as qubits,controlling the nanowires 7 or quantum structures formed therefrom,and/or taking readings from the nanowires or quantum structures formedtherefrom. For instance the gates may be arranged to tune thesemiconductor (to form dots, tunnel junctions, or appreciable electrondensity in the semiconductor) in regions without superconductor. E.g.the gates may be arranged for tuning tunnel junctions in asemiconducting network of the circuitry 8.

Some of the nanowires 7 x are not necessarily used as operative quantumcomponents and instead are simply exploited as vertical electricalcontacts, i.e. vias, between layers of circuitry 8 in the structure 6and/or to make external contact out of the top of the device 2, e.g. forconnecting to pins of the package and ultimately the external equipment.Alternatively or additionally more traditional vias may be used.

FIG. 1 illustrates for example a bottom layer of circuitry 8 b which maybe formed from the base layer of semiconductor and/or superconductor,formed directly over the substrate 5 or with only a layer of dielectricbefore meeting the substrate 5. The conductor of this layer 8 b may forexample be formed form the same superconductor and in the seamedposition step as the islands of superconductor connecting togethernanowires of the qubits (see later). FIG. 1 also shows an example toplayer or circuitry 8 t formed over the top f the filler layer. Thislayer may for example comprise contacts to the exterior of the device,such as for connecting to the external equipment (not shown). FIG. 1also illustrates one or more non-active vertical nanowires 7 x not usedfor their properties as quantum components, instead exploited as viasbetween the upper and lower layers of circuitry 8 b, 8 t. A similartechnique could be used to connect to intermediate layers of circuitrypart way through the layers 6 of the device.

Note: in some embodiments control lines or other such circuitry 8 can beformed in different layers, as each qubit requires a lot of gates andspace very quickly becomes limited.

It will be appreciated that the layers of circuitry 8 and indeed thearrangement of nanowires 7 shown in FIG. 1 are highly schematized, andare intended to illustrate the principle rather than an actual circuitor quantum structure. Note also that FIG. 1 is not intended to show thenanowires 7 as necessarily being disposed directly on the substrate 5.E.g. there may be one or more layers between the substrate 5 and thenanowires 7, e.g. a layer of dielectric.

A device 2 having the form described above can be used to implement atopological quantum computer based on Majorana zero modes (MZMs). Inoperation the chamber 2 is cooled to a temperature at which thesuperconductor material 10 in the active nanowires 7 a exhibitssuperconductivity, and the electromagnet 4 is operated to apply amagnetic field B in the direction parallel to the axis of the nanowires7 a, or at least having a substantive component in the directionparallel to the axes of the active nanowires 7 a. This induces one ormore MZMs in each of the active nanowires 7 a, enabling them forexample, to play their part as part of a MZM based qubit or network ofsuch qubits. In embodiments a pair of MZMs is induced, one at each endof the nanowire. In embodiments employing vertical nanowires 7, the MZMappearing at the bottom end (closest to the substrate 5) may beexploited to provide the desired quantum operation, e.g. as part of thequantum qubit.

Note that in the disclosed full-shell arrangement, then unlikeconventional nanowires, it is not necessary to tune the full shell wires7 into or out of the topological phase by an electrostatic potential,since the superconducting shell 10 is fully screening any electric fieldto the inside. The active full-shell wires 7 a that are to betopological instead have a reproducible growth such that the full shellwire can enter the topological phase. Given the discovery disclosedherein that full-shell nanowires support MZMs, it is possible to selectanalytically or through experimentation suitable values of B field,spin-orbit coupling, chemical potential, radius and temperature fromacross a wide range of values such that these parameters are inside thetopological phase (e.g. see FIGS. 1-4 and 16-19A and associatedworking). The inventors have performed simulations demonstrating thatthe topological phase can be obtained across a wide range of suchparameters (see also FIGS. 19B and 24B). Further, the topological phasecan be induced using only a much weaker magnetic field B than requiredin previous devices. The field is applied along the wire direction. Theminimal value depends on the radius of the wire but is typically around100 mT. This is different from half-shell nanowires where part of thenanowire is not covered by the superconductor shell and an electricfield penetrates into the semiconducting core.

It may still be involved in the full shell case is to control thecharging energy of the Majorana qubits and electrostatically change theinduced charge on the island. In order to do so only weak electricfields are required whereas in order to change the band structure oneneeds to apply very large electric fields. This refers to gating ofsuperconducting islands (e.g. by one or two electrons) connecting thenanowires 7, which may be required to control the charge of the islandsto protect form quasiparticle poisoning and enable read out (which ismuch easier than gating to change the chemical potential of thesemiconductor to induce the topological phase as in the partial-shellcase, which in any case is not possible in full-shell case as thesemiconductor is fully screened by the superconductor). The islands inquestion may for example be formed in the lower layer of circuitry 8 b,where each island connects together a set of active nanowires 7 a into aqubit (see later). This gating can be done anywhere in the island forexample by gating the full shell wires and/or the superconductingstructures (e.g. spirals) connecting the full-shell wires. The electricfield can be applied through a gate that is close the full shell wire.Where the gate is exactly does not matter. Preferably the gate shouldonly be close to a small part of the area of the full shell surface(although a larger area increases the coupling to the gate, thusreducing the required voltage changes).

FIG. 1A illustrates a semiconducting nanowire core 9 with a fullsuperconducting shell 10, subject to a weak axial magnetic field B. Thecentre region indicates the possible presence of an optional insulating(sub-)core 11 within the semiconductor core 9 (axial with thesemiconductor core 9 along its centre axis). R₁ is the radius of theouter limit of the insulating sub-core 11 (if present) from the centralaxis of the nanowire 7 (and the inner limit of the semiconductor core9). R₁ may equal 0. R₂ is the radius of the outer limit of thesemiconductor 9 core from the central axis of the nanowire 7 (and theinner limit of the superconductor coating 10). R₃ is the radius of theouter limit of the superconductor coating 10 (and of the nanowireitself).

The Hamiltonian for such a full shell nanowire with radial spin-orbitcoupling can be modelled as follows.

$H_{BdG} = {{\lbrack {\frac{p_{z}^{2}}{2m^{*}} + \frac{p_{r}^{2}}{2\; m^{*}} + \frac{( {p_{\varphi} + {e\; A_{\varphi}\tau_{z}}} )^{2}}{2\; m^{*}} - \mu} \rbrack\tau_{z}} - {\alpha_{r}\sigma_{z}{\tau_{z}( {p_{\varphi} + {e\; A_{\varphi}\tau_{z}}} )}} + {\alpha_{r}{p_{z}( {{\sigma_{y}\cos\;\varphi} - {\sigma_{x}\sin\;\varphi}} )}\tau_{z}} + {{\Delta_{0}(r)}\lbrack {{{\cos( {n\;\varphi} )}\tau_{x}} + {{\sin( {n\;\varphi} )}\tau_{y}}} \rbrack}}$The first term

$\lbrack {\frac{p_{z}^{2}}{2m^{*}} + \frac{p_{r}^{2}}{2m^{*}} + \frac{( {p_{\varphi} + {e\; A_{\varphi}\tau_{z}}} )^{2}}{2m^{*}} - \mu} \rbrack\tau_{z}$represents the kinetic energy. The middle two terms−α_(r)σ_(z)r_(z)(p_(φ)+eA_(φ)τ_(z))+α_(r)p_(z)(σ_(y) cos φ−σ_(x) sinφ)τ_(z) represent the radial spin-orbit field. The last termΔ₀(r)[cos(nφ) τ_(x)+sin(nφ)τ_(y)] represents the pairing with windingnumber n.

The system has rotational symmetry: the above BdG (Bogoliubov de Genes)Hamiltonian commutes with the following generalized angular momentumoperator:

${J_{z} = {L_{z} + {\frac{1}{2}\sigma_{z}} + {\frac{1}{2}n\;\tau_{z}}}},{\lbrack {J_{z}H_{BdG}} \rbrack = 0}$where L_(z) is the orbital angular momentum. Hence one obtains a quantumnumber labelling of angular momentum states:

$m_{J} \in \{ \begin{matrix}{\mathbb{Z}} & {{for}\mspace{14mu} n\mspace{14mu}{odd}} \\{{\mathbb{Z}} + 1} & {{for}\mspace{14mu} n\mspace{14mu}{even}}\end{matrix} $

To obtain an angle-independent Hamiltonian, angular dependence can beremoved by performing the following unitary transformation.

$H_{BdG}->{{\exp\lbrack {{- {i( {m_{J} - {\frac{1}{2}\sigma_{z}} - {\frac{1}{2}n\;\tau_{z}}} )}}\phi} \rbrack}H_{BdG}{\exp\lbrack {{i( {m_{J} - {\frac{1}{2}\sigma_{z}} - {\frac{1}{2}n\;\tau_{z}}} )}\phi} \rbrack}}$

In each m_(J) sector the Hamiltonian takes the form:

${\overset{\sim}{H}}_{BdG} = {{( {\frac{p_{z}^{2}}{2m^{*}} + \frac{p_{r}^{2}}{2m^{*}} - \mu} )\tau_{z}} + {\frac{1}{2m^{*}r^{2}}( {m_{J} - {\frac{1}{2}\sigma_{z}} - {\frac{1}{2}n\;\tau_{z}} + {\frac{e\; B\; r^{2}}{2}\tau_{z}}} )^{2}\tau_{z}} - {\frac{\alpha_{r}}{r}\sigma_{z}{\tau_{z}( {m_{J} - {\frac{1}{2}\sigma_{z}} - {\frac{1}{2}n\;\tau_{z}} + {\frac{e\; B\; r^{2}}{2}\tau_{z}}} )}} + {\alpha_{r}p_{r}\sigma_{y}\tau_{z}} + {{\Delta_{0}(r)}\tau_{x}}}$

Note that the Rashba term

$( {{{- \frac{\alpha_{r}}{r}}\sigma_{z}{\tau_{z}( {m_{J} - {\frac{1}{2}\sigma_{z}} - {\frac{1}{2}n\;\tau_{z}} + {\frac{e\; B\; r^{2}}{2}\tau_{z}}} )}} + {\alpha_{r}p_{r}\sigma_{y}\tau_{z}}} )$representing the strength of the Rashba coupling does not average out tozero. In other words the spin orbit coupling) does not cancel out.Another way to check this is to investigate angular dependence of theeigenvectors in the original basis averaging of the spin-orbit (SO) termis non-zero due to the non-trivial winding with ϕ.

FIG. 1B illustrates the thin shell limit where R₁→R₂=R. The followinggives the solution for the m_(j)=0 sector and n=1. Consider the case ofm_(j)=0 and n=1, and a thin shell of radius R consisting of a hollowsemiconductor 9 and a superconducting shell 10 as shown in FIG. 1B.

The Hamiltonian becomes simplified:

${\overset{\sim}{H}}_{BdG}^{({m_{J} = 0})} = {{( {\frac{p_{z}^{2}}{2m^{*}} - {\mu(R)}} )\tau_{z}} + {{V_{z}(R)}\sigma_{z}} + {\alpha_{r}p_{r}\sigma_{y}\tau_{z}} + {{\Delta_{0}(r)}\tau_{x}}}$where

${\mu(R)} = {{\mu - \frac{1 - n_{eff}^{2}}{8m^{*}R^{2}} - {\frac{\alpha_{r}}{2R}\mspace{14mu}{and}\mspace{14mu}{V_{Z}(R)}}} = {{n_{eff}( {\frac{1}{4m^{*}R^{2}} + \frac{\alpha_{r}}{2R}} )}.}}$μ(R) is the renormalized chemical potential, and V_(Z)(R) is theeffective Zeeman term where

$n_{eff} = {n - {\frac{\pi\; B\; R^{2}}{\Phi_{0}}.}}$The penultimate term α_(r)p_(r)σ_(y)τ_(z) is the effective Rashba SOterm, which does not commute with the Zeeman term. The last termΔ₀(r)τ_(x) is the s-wave pairing.

This Hamiltonian has all the ingredients of the standard 1D model. Thisgives the possibility of topological phases.

Note: Majorana solutions belong to the m_(J)=0 sector in order to beparticle-hole symmetric. These can only be had for n being odd.

FIG. 2 shows a simulated topological phase diagram in the shell limitfor the solution for n=1 as the winding number. The left-hand panelshows the bulk energy gap E_(gap) as a function of μ and α (the bulkenergy gap being the energy difference between the ground state and thefirst excited state of a system with no boundary such as an infinitelylong wire). The dashed line denotes the boundary of the topologicalphase in the m_(J)=0 sector, according to the above equations. Heren_(eff)=½, R/R₀=½, α₀=√{square root over (Δ₀/2 m*)} and R₀=1/√{squareroot over (2 m*Δ₀)}. For reference, using realistic parameters m*=0.026me and Δ₀=0.2 meV, one obtains α₀≈17 meV·nm and R₀=85 nm. The right-handpanel shows the bulk energy gap at fixed μ/Δ₀=2 and α/α₀=1 and afunction of n_(eff) and R. The circle in the left-hand panel marks thevalue of μ and α used in the right-hand panel, and the circle in theright-hand panel marks the values of R and ϕ used in the left-handpanel.

The dotted line indicates the topological phase transition, which occurswhen |V_(Z)(R)|=√{square root over (μ(R)²+Δ₀ ²)}. The phase diagramshows that there are stable topological regions, labelled by referencenumeral 12 in the diagram.

Consider now the full semiconducting core limit R₁=0. The Hamiltonianmay again be modelled as follows:

$H_{BdG} = {{\lbrack {\frac{p_{z}^{2}}{2m^{*}} + \frac{p_{r}^{2}}{2m^{*}} + \frac{( {p_{\varphi} + {e\; A_{\varphi}\tau_{z}}} )^{2}}{2m^{*}} - \mu} \rbrack\tau_{z}} - {\alpha_{r}\sigma_{z}{\tau_{z}( {p_{\varphi} + {e\; A_{\varphi}\tau_{z}}} )}} + {\alpha_{r}{p_{z}( {{\sigma_{y}\cos\;\varphi} - {\sigma_{x}\sin\;\varphi}} )}\tau_{z}} + {{\Delta_{0}(r)}\lbrack {{{\cos( {n\;\varphi} )}\tau_{x}} + {{\sin( {n\;\varphi} )}\tau_{y}}} \rbrack}}$where the first term

$\lbrack {\frac{p_{z}^{2}}{2m^{*}} + \frac{p_{r}^{2}}{2m^{*}} + \frac{( {p_{\varphi} + {e\; A_{\varphi}\tau_{z}}} )^{2}}{2m^{*}} - \mu} \rbrack\tau_{z}$represents the kinetic energy, the middle two terms−α_(r)σ_(z)τ_(z)(p_(φ)+eA_(φ)τ_(z))+α_(r)p_(z)(σ_(y) cos φ−σ_(x) sinφ)τ_(z) represent the radial spin-orbit field, and the last termΔ₀(r)[cos(nφ) τ_(x)+sin(nφ)τ_(y)] represents the pairing with windingnumber n.

FIG. 3(a) shows the simulated topological phase diagram for the fullcore model with R₂=100 nm, R₃=3000 nm, m*=0.026 me and Δ₀=0.2 meV. Thebulk energy gap Egap is plotted as a function of μ and α. The dashedline denotes the boundary of the topological phase obtained by findingthe zero energy crossing at p_(z)=0 in the m_(J)=0 sector. FIG. 3(b)shows band structures at the points indicated by the circles in FIG.3(a): the upper circle in FIG. 3(a) corresponds to the lower plot inFIG. 3(b), the middle circle in FIG. 3(a) corresponds to the middle plotin FIG. 3(b), and the lower circle in FIG. 3(a) corresponds to the upperplot in FIG. 3(b). In FIG. 3(b) the shade of the bands indicates whichm_(J) sector they belong to. In each plot in FIG. 3(b), the bands aboveE=0 correspond to increasing positive values of m_(J) with increasingdarkness, and the bands below E=0 correspond to decreasing negativevalues (increasing in magnitude) with increasing darkness. Theparameters are from top to bottom μ=1 meV and α={2,20,35)meV·nm.

FIG. 4 shows the simulated Majorana splitting energy and coherencelength. FIG. 4(a) shows the Majorana coherence length for the full coremodel with the same parameters as in FIG. 3. The coherence length isobtained by fitting the exponential decay of the Majorana wavefunctionin a wire of 3 μm length. The dashed line denotes the boundary of thetopological phase in the m_(J)=0 sector and the hatched region isgapless due to higher m_(J) regions. The histogram indicates thefrequency ξ of coherence lengths in the gapped topological phase. FIG.4(b) shows the Majorana wavefunction Ψ integrated over radios for μ=1meV and α=30 meV·nm. The solid line is obtained by fitting the envelopeof the wavefunction with a double exponential in the interval 0.5μm<z<2.5 μm.

As shown above, full shell nanowires support Majorana zero modes acrossa wide range of parameters, so one can use these systems for quantumcomputing purposes.

In conventional nanowires without a full coating of superconductor, theinducement of MZMs is due to the spin effect of the magnetic field B.However, in full-shell nanowires 7, the inducement of MZMs is due to acoupling between the magnetic field B and the winding of thesuperconducting phase by means of an orbital effect of the magneticfield B. A result of this is that MZMs can be induced with asignificantly lower magnitude of magnetic field. Thus this allows aquantum device or computer to work with a lower magnetic field.

The ideas outlined in Karzig et al. (Phys. Rev. B 95, 235305 (2017),“Scalable designs for quasiparticle-poisoning-protected topologicalquantum computation with Majorana zero modes”) also apply in this case.Indeed, by combining Topological (T) and non-topological (e.g.conventional s-wave (S)) segments, one can build a superconductingisland with finite charging energy and Majorana-induced ground-statedegeneracy. In addition to the planar designs explained in Karzig et al.(2017), full shell nanowires allow for vertical designs. The followingillustrates several arrangements incorporating vertical “forests” oftopological wires into qubit and computer designs. There then follows adescription of example fabrication methods and approaches which can befollowed to realized theses designs as devices.

Vertical semiconductor nanowires 7 can be used for multiple quantumcomputing applications. For example vertical nanowires 7 with full orpartial shells can host Majorana zero modes (MZM) which when combinedinto superconducting islands become qubits (tetron, 4MZMs), hexons(6MZMs), or larger quantum Hilbert spaces.

Vertical nanowires 7 with or without shells can also serve as connectorsbetween different planes (e.g. semiconductor, contact and/or gateplanes) and thus provide a valuable tool for scalable 3D integratedsolution to layout and space problems in the quantum sector of a quantumcomputer.

As another example, vertical semiconductor nanowires 7 with either fullor partial shells can also serve as conventional qubits, e.g. Gatemons,and in that capacity contribute to a solution to the 3D integrationproblem.

FIG. 5(a) illustrates an example design for a tetron qubit. FIG. 5(b)illustrates an example design for a hexon, which is a qubit plus anancilla qubit. The designs employ a vertical geometry. In either case,each tetron or hexon comprises a portion of a semiconductor network 14,which may be formed in the plane of the substrate 5. The semiconductingnetwork 14 may be formed partially or wholly from the samesemiconducting material as the nanowire cores 9, e.g. InAs or InSb. Eachtetron or hexon also comprises a superconducting island 15, which mayalso be formed in the plane of the substrate 5. The superconductingisland may be formed partially or wholly from the same superconductormaterial as used to form the nanowire coatings 10, e.g. Al or Nb. Eachtetron or hexon further comprises a plurality of vertical nanowires 7 aarranged to be invoked into the topological phase (the activenanowires), formed perpendicular to the plane of the substrate 5. Eachtetron or hexon may also optionally comprise one of more full shellnanowires 7 x disconnected from the semiconductor network 14 butconnected to a voltage source and acting as electric gates.

The tetron comprises four vertical active nanowires 7 a. The hexoncomprises six active topological nanowires 7 a. In either case, eachnanowire has one end connected to the superconducting island 15. Inembodiments, the island 15 comprises a plurality of “arms” ofsuperconductor material, the arms being connected to one another at acommon point (e.g. the centre of the island 15) but otherwise separatedfrom one another. For example in embodiments the arms may take the formof concentric spiral arms. Each active nanowire 7 a is connected at itslower end (the end closest to the substrate 5) to the distal end of adifferent respective one of the arms of the superconductor island 15,e.g. an end of a respective one of the spiral arms. The semiconductornetwork 14 connects with the superconductor island 15 at one or morepoints, for instance at one or more of the points where the activenanowires 7 a meet the superconductor island 15 (e.g. two, more or allsuch points). In embodiments this is at the distal ends of one, two,more or all of the superconductor arms.

By means of such arrangements or similar, then under conditions of amagnetic field component B perpendicular to the active nanowires 7 a, apair of MZMs 13 can be invoked in each such topological nanowire 7 a,one at each end.

The arrangement of the superconducting island 15 into thin arms (e.g.spiral arms) is useful (but not essential) as it helps prevent theformation of vortices due to quasiparticle poisoning, i.e. circulatingcurrents which may drive the superconductor 15 into the normal phase andthus hinder the formation of MZMs.

FIG. 6 illustrates a top view of an example wafer in the case of aquantum device comprising a network of hexons. The superconductor (e.g.Al) islands 15 link six MZMs 13 on the wafer to form an “insect” (withsix legs—in other words the six MZMs are connected by a central body).An additional six MZMs 13 lie vertically above at the top end of thefull shell wires 7 a. The latter are uncoupled. The spiral minimizes MZMhybridization, maximizes charging energy E_(c) in the limit of a densespiral, and allows the superconductor material 15 to be vortex free in aweak vertical B field. Vertically above each X in the diagram is atopological full shell nanowire 7 a. FIG. 7 shows further detail of anexample of how other full shell wires 7 x may be used as electric gates16 to open or close tunnel junctions. The full shells 10 meet thesuperconductor spirals 15 of the superconductor islands. The detailshows where the superconductor network 14 joins adjacent hexons andwhere, in each hexon, the semiconductor network 14 meets one of theactive vertical nanowires 7 a (where the MZMs 13 occur). The gates 16are formed from ones of the non-topological regime nanowires 7 x.

Note: with B vertical and weak, many constraints on “comb design” relax,allowing hexagonal symmetry. Note also: due to the crystal latticestructure of the superconductor e.g. aluminium), the spiral form of thesuperconductor islands 15 may not necessarily be continuously curved,but rather a series of straight edges. The term “spiral” herein does notnecessarily imply a perfectly continuous curve.

FIG. 8 shows a top view of the wafer in another example design for aquantum device comprising a network of hexons. Here the arms of thesuperconductor islands 15 do not take the form of spirals, but ratherradial arms, but otherwise the design is the same as described inrelation to FIG. 6. The radial arms functions similarly to the spiralsin that they minimize MZM hybridization, maximize charging energy E_(c),and allow the superconductor 15 to be vortex free in a weak vertical Bfield. FIG. 9 shows further detail of an example of how other full shellwires 7 x may be used as electric gates 16 to open or close tunneljunctions. The full shells 10 meet the radial superconductor arms of thesuperconductor islands 15. Again, with B vertical and weak manyconstraints on “comb design” relax, allowing hexagonal symmetry.

FIG. 10 shows a top-down view of a square layout of tetrons, similar tothe hexon arrangement of FIGS. 6 and 7 but with tetrons comprising onlyfour active vertical nanowires per superconductor island 15 instead ofsix, and the tetrons being linked by square or octagonal cells of thesemiconductor 14 rather than a hexagonal comb. Each superconductorisland (e.g. Al) 15 supports four plus four MZMs (one at each end ofeach active vertical nanowire 7 a). The tunnel junction in thesemiconductor network 14 may be gate controlled by further vertical fullshell nanowires 7 x as shown in the detail of FIGS. 7 and 9. Anadvantage of the design of FIG. 10 is slightly smaller superconductorislands 15, and a higher charging energy E_(c).

FIG. 11 shows another top-down view of a square layout of tetrons,similar to the tetron arrangement of FIG. 10, but with radial instead ofspiral arms 15 as in FIG. 8. Again the tunnel junction in thesemiconductor network 14 may be gate controlled by further vertical fullshell nanowires 7 x as shown in the detail of FIGS. 7 and 9. Again anadvantage of the design of FIG. 10 is slightly smaller superconductorislands 15, and a higher charging energy E_(c).

FIG. 12 illustrates some example fabrication steps that may be used toform devices having the structures disclosed herein, such as those ofFIG. 1 to 1B or 5 to 11. FIG. 12 includes fabrication steps to produceselective area grown (SAG) or vapour-liquid-solid (VLS) superconductoron ridges or cliffs for the disclosed designs.

Step i) begins with a substrate 5 and a dielectric layer 17 formed overthe substrate 5, with etched windows 22 formed through the dielectric 17for selective area growth (SAG). These windows 22 are placed where thevertical nanowires 7 are to be grown (one window 22 for each suchnanowire 7). The dielectric 17 may be any suitable dielectric insulator.The etching of the windows may be performed using any suitable knownetching process.

The method proceeds from step i) to step ii) or ii′). Either way, inthis next step the substrate 5 is etched with trenches 23. Any suitableknown etching process may be used. In variant ii) a further layer ofdielectric 17 is applied which covers the inside surfaces of thetrenches 23. In variant ii)′ a further layer of dielectric 17 may beapplied, but if so then it does not cover the trenches 23. Any suitableselective deposition technique may be used to apply the further layer ofdielectric, e.g. using a mask. The dielectric material 17 applied instep ii) or ii′) may be the same as applied in step i).

The method then proceeds from step ii) to step iii) or from step ii′) tostep iii′). In either variant, here a portion of semiconductor material18 is formed in the window 22 that was formed for each vertical nanowire7 in step i). This semiconductor 18 is deposited using selective areagrowth (SAG). In variant iii′) the semiconductor material 18 may also bedeposited in inside surfaces of the trenches 23. The same semiconductormaterial 18 may be used as that used to fill the windows, and SAG mayalso be used. The portions of semiconductor 18 in the windows 22 andtrenches 23 may be formed in the same SAG step.

The method then proceeds from step iii) to step iv) or from step iii′)to step iv′). In either variant this step comprises growing thesemiconductor cores 9 of the nanowires 7 from a semiconductor material18, using VLS to grow the cores 9 of the wires vertically under SAGconditions. The semiconductor material 18 used to form the nanowirecores 9 in this manner may be the same as that used in step iii or iii′)respectively, e.g. InSb or InAs.

The method then proceeds from step iv) to step v) or from step iv′) tostep v′). In either variant this step comprises depositing asuperconductor material 19 over the dielectric 17 and the core 9 of thevertical nanowires 7. The superconductor material 19 may for example beAl or Nb. The deposition may be done though angle deposition of thesuperconductor material 19 with a grazing angle. FIG. 12v ′) shows abeam flux of superconducting material. The beam flux may also be used inv) but is not shown for simplicity of illustration. In order to get thewire fully covered, the beam or wafer may be rotated (while keeping thegrazing angle in order not to fill the trenches). Depending on angle,the superconductor 19 may or may not meet the semiconductor 18 ifpresent in the trenches 23. The superconductor material 19 that formsover the cores 9 of the nanowires 7 forms the coating 10. Either thewafer or the beam may be rotated in order to achieve a full shell 10around the vertical nanowires 7. Regions of the superconductor material19 formed in the plane of the substrate 5 may be used to form thesuperconducting islands 15.

Depending on application it may be desired to grow semiconductor 18 inthe trenches 23 (e.g. to make quantum dots protected from superconductordeposition) or to keep the trenches 23 free of semiconductor growth(e.g. to provide better electrical insulation). These are the unprimed(ii-v) and primed (ii′-v′) cases respectively.

The semiconductor 18 and superconductor 19 other than that used to formthe cores 9 and coating 10 of the vertical nanowires 7, respectively,may be used to form part or all of one of the layers of circuitry 8,such as the bottom layer of circuitry 8 b. As mentioned, this couldinclude the superconducting islands 15, quantum dots in semiconductor18, other parts of the semiconductor network 14, gates 16, contacts forgating, and/or other contacts (e.g. external contacts). Alternatively oradditionally, part of all of some or all of these components, and/orother components, may be formed in one or more subsequent steps in oneor more other layers 6 formed over the substrate 5.

FIG. 13 illustrates an isometric view of one example design for allbottom layer control. Here, any arbitrary pattern can be etched in thesuperconductor 19 in the SAG bottom plane. The dotted circle shows aside view of the semiconductor layer 18 between the substrate 5 and thesuperconductor layer 19. The nanowire cores 9 may be formed using VLS.The other regions of the semiconductor 18 in the plane of the substrate5 may be formed using SAG. The dielectric layer 17 is not shown in FIG.13 but it will be understood that it may be present in embodiments.

FIG. 14 illustrates some further design features that may be employed inembodiments. FIG. 14(a) shows a side view of the wafer, in which afiller material 20 has been deposited over the semiconductor layer 18,superconductor layer 19 and any other parts of the bottom layer ofcircuitry 8 b. The filler material 20 may for example be a wax, or aplastic such as PMMA. It serves to mechanically support the verticalnanowires 7 in the final, finished device 2. The bottom layer ofcircuitry 8 b may include one or more parts (e.g. semiconductor)suitable for making one or more external contacts 21 b with the externalequipment (not shown). A top layer of circuitry 8 t may be formed overthe filler material 20. The top layer of circuitry 18 t may include oneor more parts (e.g. superconductor 19) suitable for making one or moreexternal contacts 21 t with the external equipment. FIG. 14(b) shows asimilar arrangement that alternatively or additionally includes at leastone intermediate layer of circuitry 8 i, beneath and over which thefiller 20 is formed. The intermediate layer of circuitry 8 i may includeone or more parts (e.g. superconductor 19) for making one or moreexternal contacts 21 i with the external equipment. The externalcontacts 21 formed from the bottom, intermediate and/or top layers ofcircuitry 8 may for example be used to control the qubits (such as forgating), and/or to take measurements, and/or supply power to power thedevice 2. For instance in embodiments the intermediate layer ofcircuitry 8 i may comprise a gating plane. FIG. 14(c) shows an isometricview of another example, in this case where a contact 24 of a gate 16 isformed in the top layer of circuitry 8 t over the filler 20. Thiscontact 24 can be used to apply a potential controlled by the externalequipment for gating purposes. Alternatively the gate contact 24 couldbe formed amongst the filler 20 in an intermediate layer 8 i or bottomlayer 8 b of circuitry 8.

Further Discussion

As described above, the present disclosure describes a new model systemsupporting Majorana zero modes based on semiconductor nanowires with afull superconducting shell. It is shown that, in the presence of aradial spin-orbit field, the winding of the superconducting orderparameter is sufficient to drive the system into a topological phase,thus dramatically reducing the magnitude of the required magnetic field.The resulting topological phase persists over a large range of chemicalpotentials, opening the way to the realization of Majorana zero modes inan experimental system of easy fabrication and reproducible materialquality.

Majorana zero modes (MZMs) hold the promise to revolutionize quantumcomputation through topological quantum information processing. In thelast decade, research in MZMs evolved into one of the most active fieldsin modern condensed matter physics, with astonishing progress in theirexperimental realization and theoretical understanding. Much of thisactivity was fueled by proposals of simple, experimentally viablesystems. A particularly promising route involves proximitizedsemiconducting nanowires (e.g. see Lutchyn et al, “Majorana Fermions anda Topological Phase Transition in Semiconductor-SuperconductorHeterostructures”, Physical Review Letters 105, 077001 (2010), 13 Aug.2010)). The essential ingredients in these schemes are quite simple:spin-orbit coupling, a Zeeman field, and induced superconductivity.Nevertheless, the required coexistence of large (˜1 T) magnetic fieldswith superconductivity, as well as the need for careful control of thechemical potential in the semiconductor, pose important challengestowards a consistent realization of MZMs in nanofabricated devices,requiring ongoing experimental improvements.

In the present disclosure, there are described semiconducting nanowiresfully covered by a superconducting shell as an alternative structure torealize MZMs. While being of similar simplicity and practicalfeasibility as the original nanowire proposals (e.g. of Lutchyn et al),full-shell nanowires can be used to provide any one or more of a numberof key advantages. First and foremost, the topological transition in afull shell wire is driven by the field-induced winding of thesuperconducting order parameter, rather than by the Zeeman effect, andso the required magnetic fields can be very low (˜0.1 T). Moreover, thefull shell naturally protects the semiconductor from impurities andrandom surface doping, thus enabling a reproducible way of growing manywires with essentially identical properties. Although full-shell wiresdo not allow for direct electrostatic gating of the electron density inthe semiconducting core, it is demonstrated below that via a carefuldesign of the wire properties, e.g. by choosing the right radius, it ispossible to obtain wires that naturally harbor MZMs at certain magneticfield.

While it is known that well-chosen superconducting phase differences orvortices can be used to break time-reversal symmetry and localize MZMsin topological insulators and semiconductor heterostructures, thecorresponding realizations typically require careful tuning of thefluxes which would complicate a scalable approach with multiple MZMs(see the above-reference paper by Karzig et al). Here, it is shown thatthe quantized superconducting winding number in a full-shell wire is anatural and more robust implementation of the wanted phase differences,leading to sizable regions of topological phase space. Below, firstthese ideas are demonstrated in a simple model of a hollow wire, wherean analytic mapping to the standard model of a topologicalsuperconductor is possible. These results are then complemented withnumerical and analytical studies of the topological phase in theopposite regime where the electron density is spread out over the entiresemiconducting core.

Theoretical model: consider a nanowire consisting of a semiconducting(SM) core and a full superconducting (SC) shell, as illustrated in FIG.1A. This Illustrates a semiconducting nanowire core 9 with a fullsuperconducting shell 10, subject to a weak axial magnetic field B.Bottom left. In the detail of the cross-section, the region with r<R₁indicates the possible presence of an insulating core 11 in thesemiconductor. FIG. 15 is a sketch of the energy gap in thesuperconducting shell as a function of the magnetic field, exhibitingcharacteristic Little-Parks lobes. Different lobes correspond todifferent winding numbers n of the superconducting order parameteraround the wire. The period is B₀≈4Φ₀/π(R₂+R₃)2 with Φ₀=h/2e thesuperconducting flux quantum.

Assume that the semiconductor (e.g. InAs) has a large Rashba spin-orbitcoupling due to an intrinsic electric field at thesemiconductor-superconductor interface. The system is subject to amagnetic field along the direction of the nanowire {circumflex over(z)}, i.e. {tilde over (B)}=B{circumflex over (z)}. Using cylindricalcoordinates and the symmetric gauge for the electromagnetic vectorpotential,

${\overset{->}{A} = {\frac{1}{2}( {\overset{->}{B} \times \overset{->}{r}} )}},$the effective Hamiltonian for the semiconducting core can be written as(here ℏ=1):

$\begin{matrix}{H_{0} = {\frac{( {\overset{->}{p} + {e\; A_{\varphi}\hat{\varphi}}} )^{2}}{2\; m^{*}} - \mu + {\alpha\;{\hat{r} \cdot \lbrack {\overset{->}{\sigma} \times ( {\overset{->}{p} + {e\; A_{\varphi}\hat{\varphi}}} )} \rbrack}}}} & (1)\end{matrix}$

Here {right arrow over (p)} is the electron momentum, e>0 the electriccharge, m* the effective mass, μ is the chemical potential, α thestrength of the radial spin-orbit coupling, and finally σi are spin-½Pauli matrices. Note that both μ and α may depend on the radialcoordinate r. The vector potential Δ_(ϕ)=Φ(r)/2πr, where Φ(r)=πBr² isthe flux threading the cross-section at radius r. For simplicity, weneglect the Zeeman term due to the small magnetic fields required inthese devices.

The shell (e.g. made out of Al) induces superconducting correlations inthe nanowire due to Andreev processes at thesemiconductor-superconductor (SM-SC) interface. If the coupling to thesuperconductor is weak, the induced pairing in the nanowire can beexpressed as a local potential Δ({right arrow over (r)}), see AppendixB. In the Nambu basis Ψ=(ψ↑, ψ↓, ψ_(↓) ^(†), −ψ_(↑) ^(†)), theBogoliubov-de-Gennes (BdG) Hamiltonian for the proximitized nanowire isthen given by:

$\begin{matrix}{H_{B\; d\; G} = \begin{bmatrix}{H_{0}( \overset{->}{A} )} & {\Delta( \overset{->}{r} )} \\{\Delta^{*}( \overset{->}{r} )} & {{- \sigma_{y}}{H_{0}( {- \overset{->}{A}} )}^{*}\sigma_{y}}\end{bmatrix}} & (2)\end{matrix}$

Assume that the thickness of the SC shell is smaller than Londonpenetration depth: R₃−R₂<<λ_(L). Therefore, the magnetic flux threadingthe SC shell is not quantized. However, the magnetic field induces thewinding of the superconducting phase, i.e. the order parameter Δ({rightarrow over (r)})=Δ(r)e^(−inφ) with φ the angular coordinate and n∈

the winding number. In practice, the winding number n adjusts itself tothe value of the external magnetic field so that the free energy of thesuperconducting shell is minimized. This is the familiar Little-Parkseffect: the changes in winding number lead to periodic lobes in theenergy spectrum of the superconducting shell, see FIG. 15 and AppendixA.

Notice the following rotational symmetry of the BdG Hamiltonian:

$\begin{matrix}{{\lbrack {J_{z},H_{B\; d\; G}} \rbrack = 0},{{{with}\mspace{14mu} J_{z}} = {{{- i}\;{\partial_{\varphi}{+ \frac{1}{2}}}\sigma_{z}} + {\frac{1}{2}n\;\tau_{x}}}},} & (3)\end{matrix}$where τ_(i) matrices acting in Nambu space are introduced. Eigenstatesof H_(BdG) can thus be labeled by a conserved quantum number m_(i):

$\begin{matrix}{{\Psi_{m_{J}}( {r,\varphi,z} )} \propto {e^{{i{({m_{J} - {\frac{1}{2}\sigma_{z}} - {\frac{1}{2}n\; r_{z}}})}}\varphi}{{\Psi_{m_{J}}( {r,z} )}.}}} & (4)\end{matrix}$

The wave function has to be single-valued, which imposes the followingconstraint on m_(J):

$\begin{matrix}{m_{J} \in \{ \begin{matrix}{\mathbb{Z}} & {{n\mspace{14mu}{odd}},} \\{{\mathbb{Z}} + \frac{1}{2}} & {n\mspace{14mu}{{even}.}}\end{matrix} } & (5)\end{matrix}$

Now remove the angular dependence of H_(BdG) via a unitarytransformation

${U = {\exp\lbrack {{- {i( {m_{J} - {\frac{1}{2}\sigma_{z}} - {\frac{1}{2}n\;\tau_{z}}} )}}\varphi} \rbrack}},$namely {tilde over (H)}_(BdG)=UH_(BdG)U^(†) where:

$\begin{matrix}{{\overset{\sim}{H}}_{B\; d\; G} = {{( {\frac{p_{z}^{2}}{2\; m^{*}} + \frac{p_{r}^{2}}{2\; m^{*}} - \mu} )\tau_{z}} + {\frac{1}{2\; m^{*}r^{2}}( {m_{J} - {\frac{1}{2}\sigma_{z}} - {\frac{1}{2}n\;\tau_{z}} + {e\; A_{\varphi}r\;\tau_{z}}} )^{2}\tau_{x}} - {\frac{\alpha}{r}\sigma_{z}{\tau_{z}( {m_{J} - {\frac{1}{2}\sigma_{z}} - {\frac{1}{2}n\;\tau_{z}} + {e\; A_{\varphi}r\;\tau_{z}}} )}} + {\alpha\; p_{z}\sigma_{y}\tau_{z}} + {{\Delta(r)}{\tau_{x}.}}}} & (6)\end{matrix}$

Note that naively one would expect spin-orbit coupling to average out.However, the non-trivial structure of m_(J) eigenvectors yields finitematrix elements proportional to the Rashba spin-orbit coupling.

It will now be shown that the above BdG Hamiltonian supports topologicalsuperconductivity and Majorana zero modes. First, notice thatparticle-hole symmetry relates states with opposite energy and angularquantum number m_(J), i.e., PΨ_(E,mJ)=Ψ_(−E,−mJ) with P=τ_(y)σ_(y)K,where K represents complex conjugation. Thus, a non-degenerate Majoranazero-energy solution can only appear in the m_(J)=0 sector. In turn,this implies that Majorana zero modes can appear only when the windingnumber n is odd.

Hollow cylinder model: now focus on the limit in which the semiconductorforms a thin-wall hollow cylinder (i.e., R₁≈R₂ in FIGS. 1A and 15). Thisapproximation is motivated by the fact that there is an accumulationlayer in some semiconductor-superconductor heterostructures such asInAs/Al, so that the electron density is concentrated within thescreening length (˜20 nm) from the interface. In this case, one canconsider only the lowest-energy radial mode in Equation (6). This allowsan analytical solution of the model. If one considers the sector m_(J)=0for n=1, this arrives at the simplified Hamiltonian:

$\begin{matrix}{{\overset{\sim}{H}}_{0} = {{( {\frac{p_{z}^{2}}{2\; m^{*}} - \mu_{0}} )\tau_{z}} + {V_{Z}\sigma_{z}} + {\alpha\; p_{z}\sigma_{y}\tau_{x}} + {\Delta\;\tau_{x}}}} & (7)\end{matrix}$

Here, Δ≡Δ(R₂), and the effective chemical potential and Zeeman energy isdefined as:

$\begin{matrix}{\mu_{0} = {{\mu - \frac{1 + \lbrack {1 - {{\Phi( R_{2} )}/\Phi_{0}}} \rbrack^{2}}{8\; m^{*}R_{2}^{2}} - {\frac{\alpha}{2\; R_{2}}V_{Z}}} = {\lbrack {1 - \frac{\Phi( R_{2} )}{\Phi_{0}}} \rbrack{( {\frac{1}{4\; m^{*}R_{2}^{2}} + \frac{\alpha}{2\; R_{2}}} ).}}}} & (8)\end{matrix}$

Notice that when the core is penetrated by one flux quantum, i.e.Φ(R₂)=Φ₀, V_(Z)=0. This regime corresponds to the trivial (i.e. s-wave)superconducting phase. However, a small deviation of the magnetic fieldcan drive the system into the topological phase (note that magnetic fluxpiercing the finite-thickness superconducting shell can be significantlydifferent from that penetrating through the semiconducting shell).Indeed, the Zeeman and spin-orbit terms in Equation (7) do not commuteand thus V_(Z) opens a gap in the spectrum at p_(z)=0.

One can map Equation (7) to a Majorana nanowire model (e.g. the model ofthe above-cited paper by Lutchyn et al) and determine the topologicalphase diagram by calculating the topological index Q. The topologicalquantum phase transition between trivial (Q=1) and non-trivial (Q=−1)superconducting phases occurs when:|V _(Z)|=√{square root over (μ₀ ²+Δ²)}  (9)

The resulting phase diagram is shown in FIG. 16, where the gap closingat the topological transition in the m_(J)=0 is indicated by the dashedlines. FIG. 16 shows the topological phase diagram of the hollowcylinder model. Panel (a) shows the bulk energy gap E_(gap) as afunction of chemical potential and spin-orbit coupling. The energy gapis multiplied by the topological index Q=±1. The dashed line denotes theboundary of the topological phase in the m_(J)=0 sector, as denoted bythe written labels and according to Equation (9). Here,

${\frac{\Phi( R_{2} )}{\Phi_{0}} = \frac{1}{2}},{\frac{R}{R_{0}} = \frac{1}{2}},$as indicated by a black star in panel (b). It is defined thatα₀=√{square root over (Δ/2 m)} and R₀=1/√{square root over (2 mΔ)}. Forreference, using realistic parameters m*=0.026 m_(c) and Δ=0.2 meV, oneobtains α₀≈17 meV·nm and R₀=85 nm. Right: Bulk energy gap at fixed μ/Δ=2and α/α₀=1, as indicated by the black star in (a), as a function of fluxand R. Panels (c-e) show band-structures at the points indicated in (a)by black triangles, illustrating the closing and re-opening of the bulkgap in the m_(J)=0 sector.

Close to the transition, the quasiparticle spectrum is determined bym_(J)=0 sector and is given by:

$\begin{matrix}{{E( p_{z} )} = {\sqrt{V_{Z}^{2} + \Delta^{2} + \mu_{0}^{2} + {\alpha^{2}p_{z}^{2}} - {2\sqrt{{\alpha^{2}p_{z}^{2}\mu_{0}^{2}} + {V_{Z}^{2}( {\Delta^{2} + \mu_{0}^{2}} )}}}}.}} & (10)\end{matrix}$

One can estimate quasiparticle velocity v using Equation 10 to findv=αΔ√(Δ²+μ₀ ²) and corresponding coherence length ξ˜v/E_(gap) whereE_(gap) is the quasiparticle gap in m_(J)=0 sector.

A well-defined topological phase requires the quasi-particle bulk gap tobe finite in all angular momentum channels, and not only for m_(J)=0.Therefore, in FIG. 16 there is shown the energy gap of the modeldetermined by taking into account all m_(J) sectors. This restricts theextent of the topological phase, since at large chemical potentialsand/or spin-orbit couplings the energy gap vanishes in sectors withhigher m_(J)(see Appendix. C). Nevertheless, FIG. 16 demonstrates thatin the hollow cylinder model a gapped topological phase exists over afinite range in all the model parameters, with optimal quasiparticlegaps comparable in magnitude to Δ and the corresponding coherence lengthξ100 nm.

Full cylinder model: now consider the case in which the electron densityis uniform in the semiconducting core (i.e., R₁=0 in FIGS. 1A and 15).One can solve for the radial modes in the core numerically for thedifferent m_(J) quantum numbers. The superconductor is treatedeffectively as a boundary condition at r=R₂, neglecting the effect ofthe magnetic field penetrating the shell, so that Φ(R₃)≈Φ(R₂). Thistreatment of the proximity effect is justified for a thinsuperconducting shell in the dirty limit (see Appendixes A-D fortechnical details).

FIG. 17 shows the topological phase diagram for the full cylinder modelwith parameters appropriate for InAs—Al hybrid semi-superconductornanowires. Panel (a) shows the topological phase diagram for the fullcylinder model with R₂=100 nm, m*=0.026 m_(e), Δ=0.2 meV, andΦ(R₂)=Φ₀/2. Panel (b) shows the same but for Φ(R₂)=Φ₀. The bulk energygap E_(gap), multiplied by the topological index

is plotted as a function of μ and α. The dashed line denotes theboundary of the topological phase obtained by finding the zero energycrossing at p_(z)=0 in the m_(J)=0 sector. Panels (c-f) show bandstructures at the points indicated in (b). The parameters are from leftto right μ=1 meV and α={2,20,30,35}meVnm.

As in the hollow cylinder model, one finds a stable topological phasewhich extends over reasonably large range of the chemical potential andhas the maximum topological gap of order 30 μeV. A large part of thetopological phase is gapless due to m_(J)≠6 states as in the previouscase. Due to the large extend of the radial wavefunction into thesemiconducting core the topological gap is smaller in full cylindermodel. Also, the cancellation of superconducting winding by the orbitaleffect is not exact in this case, so that a topological phase alsoappears at Φ(R₂)=Φ₀ for appropriate parameters, see also Appendix E. InFIG. 17(b-e) there is shown the momentum dispersion of different m_(J)sectors, illustrating the topological transition in and out of thegapped topological phase. The bands forming in the core of the wire havea distinctly flat dispersion as can be seen in FIG. 17(b), which isreminiscent of Caroli-de Gennes-Matricon vortex states.

The evolution of the local density of states (DOS) at the end of afinite wire as a function of magnetic field is shown in FIG. 18(a).Panel (a) shoes DOS at the end of the wire as a function of flux forμ=1.1 meV, α=30 meVnm and the same parameters as in FIG. 17(b) Majoranacoherence length for the full cylinder model with the same parameters asin FIG. 178(b) with Φ(R₂)=Φ₀. The coherence length is obtained byfitting the exponential decay of the Majorana wavefunction in a wire of3 μm length. The dashed line denotes the boundary of the topologicalphase in the m_(J)=0 sector and the red shaded region is gapless due tohigher m_(J) sectors. The inset shows the Majorana wavefunctionintegrated over radius for μ=1 meV and α=30 meVnm (this point is alsoindicated by the black dot). The solid red line is obtained by fittingthe envelope of the wavefunction with a double exponential in theinterval 0.5 μm<z<2.5 μm. The histogram indicates how often a givencoherence length is found inside of the gapped topological phase. Panel(b) shows DOS at the end of the wire as a function of flux for μ=1.1 meVand α=30 meVnm.

As the flux Φ(R₂) is increased, the winding number n changes by one atevery half-integer multiple of Φ₀. The change in winding number causes adiscontinuous jump in the density of states. At energies comparable tothe pairing gap, the DOS reproduces the periodic Little-Parks lobesalready sketched in FIG. 15. However, at lower energies, the DOS revealsthe different structure of the sub-gap spectrum in the semiconductingcore. A peak in the DOS is visible at zero energy in the n=1 and n=3lobe, but not in the n=0 and n=2 lobes, in agreement with the fact thatMajorana zero modes should only appear for odd values of n. Within oddlobes one can see the characteristic asymmetry of the subgap spectrawith respect to the center of the lobe which stems from the differencein magnetic flux penetrating the core of the wire and thesuperconducting shell. The disappearance of the Majorana zero-energystates within odd lobes occurs because both μ₀ and V_(Z) depend on themagnetic field, see Equations (8) which, at some point, leads to thetopological phase transition. The bulk gap closing is clearly visiblewhen plotting the DOS in the middle of the wire, see Appendix F.

In FIG. 18(b), there is investigated Majorana hybridization due to afinite nanowire length (L 3 μm) and extract the coherence length byfitting the Majorana wave function, see the inset. Despite therelatively small topological gaps of FIG. 17, one finds quite shortMajorana coherence lengths with the minimum being of the order of 250 nmwhich is related to the small group velocity of bulk states of FIG.17(c).

Consider now perturbations breaking the angular symmetry discussedabove. Such perturbations (e.g., disorder in the superconducting shellor other imperfections) are ubiquitous in realistic devices and wouldcouple different m_(J) eigenstates. This may actually have beneficialconsequences for the stability of the topological phase since couplingdifferent m_(J) sectors may induce a gap in the gapless regionsoccurring at large p and a in FIG. 17. As a result, such perturbationsmay lead to a significantly larger topological phase space volume. Theinventors have considered effect of the disorder in the superconductorand confirmed this conjecture, see FIG. 19A.

Panel (a) in FIG. 19A shows a topological phase diagram for thedisordered full cylinder model with the same parameters as in FIG. 17.To allow for cylindrical symmetry breaking perturbations the Hamiltonian(2) is discretized on a square lattice with α=10 nm in the twodimensional cross section. The disorder potential δU on each latticesite in the superconductor is uniformly distributed δU∈[−U,U] with U=2meV. The gap times the topological index is averaged over twelvedisorder configurations. The red dashed line indicates the phaseboundary without disorder. Panel (b) shows band structures at α=35 eVnmand μ=1.5 meV for increasing disorder strength from top to bottomU={0,2,4}meV. A single disorder configuration is shown.

In fact, the inventors have found that the phase space for thetopological phase in full-shell nanowires is very large, supporting MZMsover a wide range of parameters (B field, spin-orbit coupling, chemicalpotential, radius and temperature). This is demonstrated in FIG. 19B.

FIG. 19B shows the results of an investigation into the effects ofangular-symmetry-breaking perturbations on the phase diagram. Suchperturbations (e.g., shape deformations or the disorder in thesuperconducting shell) are ubiquitous in realistic devices and wouldcouple different m_(J) eigenstates which can have beneficialconsequences for the stability of the topological phase. Indeed, theperturbations that couple different m_(J) sectors may open a gap in theregions of large μ and |α|. Hatched regions are topological. Panel (a)shows the topological phase diagram for the hollow cylinder model withbroken angular symmetry. The symmetry is broken by introducing ananisotropic effective mass that is dependent on φ in the z-direction:1/m*₂=1/m*(1+q cos(2φ). Results are shown for q=1, Φ(R₂)/Φ₀=½ andR/R₀=½.

Panel (b) of FIG. 19B shows the topological phase diagram for thedisordered full cylinder model with the same parameters as in FIG. 3(a)using Φ(R₂)/Φ₀=1. To allow for cylindrical symmetry breakingperturbations the Hamiltonian is discretized on a square lattice withα=10 nm in the two-dimensional cross section. The disorder potential ΔUon each lattice site in the superconductor is uniformly distributedΔU∈[−U, U] with U=2 meV. The scenario of FIG. 19B considers rotationalsymmetry breaking disorder that is translationally invariant in the zdirection. The gap times the topological index is averaged over twelverealizations. The dashed lines indicate the phase boundaries withoutdisorder. Below are shown the band structures at α=35 eV nm and μ=1.5meV for increasing disorder strength from left to right U={0, 2, 4} meV.A single disorder configuration is shown.

To summarize, the above has investigated a novel physical systemsupporting Majorana zero modes based on semiconductor nanowires coveredby a superconducting shell. Using a combination of analytical andnumerical methods, it has been calculated the topological phase diagramand show that the model supports robust topological superconductivity ina reasonably large parameter space. The topological phase can becharacterized by calculating quasiparticle gap and effective coherencelength. The existence of a readily accessible robust topological phasein full-shell nanowires opens a pathway for realizing topologicalquantum computing devices.

Appendix A: Model for the Disordered Superconducting Shell

In this section, there is considered a disordered superconducting shell(e.g. Al shell) with inner and outer radii R₂ and R₃, respectively, seeFIGS. 1A and 15. It is assumed that the thickness of the shelld≡R₃−R₂<<λ_(L), with λ_(L) being the London penetration length in thebulk superconductor. In this case, the screening of the magnetic fieldby the superconductor is weak and can be neglected. The effectiveHamiltonian for the SC shell in cylindrical coordinates can be writtenas:

$\begin{matrix}{H_{B\; d\; G}^{(s)} = {{\lbrack {\frac{{\hat{p}}_{z}^{2}}{2\; m^{*}} + \frac{{\hat{p}}_{r}^{2}}{2\; m^{*}} + \frac{( {{\overset{\sim}{p}}_{\varphi} + {e\; A_{\varphi}\tau_{z}}} )^{2}}{2\; m^{*}} - \mu^{(s)} + V_{i\;{mp}}} \rbrack\tau_{z}} + {\Delta_{0}\lbrack {{{\cos( {n\;\varphi} )}\tau_{x}} + {{\sin( {n\;\varphi} )}\tau_{y}}} }}} & ({A1})\end{matrix}$

Here, {circumflex over (p)}_(i) are the electron momentum operators, e>0the electric charge, m the electron mass in the SC,

${A_{\varphi} = {\frac{1}{2}B\; r}},\mu^{(s)}$is the chemical potential in the SC, τ_(i) are Pauli matricesrepresenting Nambu space, Δ₀ is bulk SC gap at B=0, n is the windingnumber for the SC phase, and V_(imp) represents short-range impurityscattering potential. It is enlightening to perform a gaugetransformation which results in a real order parameter, i.e.Δ₀[cos(nϕ)τ_(x)+sin(nϕ)τ_(y)]→Δ₀τ_(x). The gauge transformationintroduces an effective vector potential, Δ_(φ)=Ã_(φ) with:

$\begin{matrix}{{\overset{\sim}{A}}_{\varphi} = {{{- \frac{1}{2e\; r}}( {n - {2\; e\; A_{\varphi}r}} )} = {- {\frac{1}{2e\; r}\lbrack {n - \frac{\Phi(r)}{\Phi_{0}}} \rbrack}}}} & ({A2})\end{matrix}$where Φ(r)=πBr2 and Φ0=h/2e. It follows from this argument that thesolution of Equation (A1) should be periodic with Φ₀, see FIG. 20.Namely, the winding number adjusts itself to the value of the magneticfield so that the energy of the superconductor is minimized. Inparticular, for each winding number n, the maxima of the quasiparticlegap occur at:

$\begin{matrix}{B_{n} \approx {4n\frac{\Phi_{0}}{{\pi( {R_{2} + R_{3}} )}^{2}}}} & ({A3})\end{matrix}$

The Zeeman contribution can be neglected since the typical magneticfields of interest are smaller than 100 mT for which the Zeemansplitting is negligible.

FIG. 20 shows a simulation of a superconducting shell, without thesemiconducting core, with R₁=R₂=60 nm and R₃=70 nm. Realistic parameterscorresponding to Al are used: m*=m_(e), μ=10 eV and Δ₀=0.34 meV [32].The Hamiltonian Equation (A1) is discretized on a square lattice withα=0.1 nm using the Kwant package. In (a) we show the clean case, wherethe superconductivity is destroyed almost immediately by the magneticfield. In (b) we show the disordered case using the on-site disorderpotential δU which is randomly sampled from δU∈[−U,U] with U=2 eV. Thedisorder is applied in an outer layer of 5 nm thickness, with thepurpose of modelling an oxidized Al₂O₃ layer.

In order to understand the magnetic field dependence of thequasiparticle gap, one needs to calculate the Green's functions for thedisordered SC shell as a function of Ã_(φ). The disorderedsuperconductor is characterized by an elastic mean free path l_(e) and acorresponding diffusive coherence length ξ_(d)=√{square root over(l_(e)ξ₀>>)}l_(e), where ξ₀=v_(F)/Δ is the coherence length in the bulk,clean limit (v_(F) is the Fermi velocity in the SC). For simplicity, weassume henceforth that the thickness of the superconducting shell d &ξ_(d), so that the properties of the system can be obtained bycalculating the Green's function for the disordered bulk superconductorin magnetic field B and n=0. This problem was considered by Larkin, whoshowed that within the self-consistent Born approximation the normal andanomalous Matsubara Green's function are given by:

$\begin{matrix}{{G^{(m_{J})}( {\omega_{n},ɛ} )} = \frac{{i\;\omega_{n}} + \overset{\_}{G} + {H\; m_{J}}}{( {\Delta + \overset{\_}{F}} )^{2} + ɛ^{2} - ( {{i\;\omega_{n}} + \overset{\_}{G} + {H\; m_{J}}} )^{2}}} & ({A4}) \\{{F^{(m_{J})}( {\omega_{n},ɛ} )} = {- \frac{\Delta + \overset{\_}{F}}{( {\Delta + \overset{\_}{F}} )^{2} + ɛ^{2} - ( {{i\;\omega_{n}} + \overset{\_}{G} + {H\; m_{J}}} )^{2}}}} & ({A5})\end{matrix}$where H=eB/4 m and m_(J) is the angular momentum eigenvalue and is theeigenvalue of the Hamiltonian:H₀ ^(SC)ϕ({right arrow over (r)})=εϕ({right arrow over (r)}), where

$H_{0}^{SC} = {\frac{{\hat{p}}_{z}^{2}}{2m^{*}} + \frac{{\hat{p}}_{r}^{2}}{2m^{*}} + \frac{{\hat{p}}_{\varphi}^{2}}{2m^{*}} - \mu^{(s)}}$

The functions G and F are determined by the following equations:

$\begin{matrix}{\overset{\_}{G} = {\frac{1}{2\tau\;{\overset{\_}{m}}_{J}}{\sum\limits_{{m_{J}} < {\overset{\_}{m}}_{J}}\;\frac{{i\;\omega_{n}} + \overset{\_}{G} + {H\; m_{J}}}{\sqrt{( {\Delta + \overset{\_}{F}} )^{2} - ( {{i\;\omega_{n}} + \overset{\_}{G} + {H\; m_{J}}} )^{2}}}}}} & ({A6}) \\{\overset{\_}{F} = {\frac{1}{2\tau\;{\overset{\_}{m}}_{J}}{\sum\limits_{{m_{J}} < {\overset{\_}{m}}_{J}}\;\frac{\Delta + \overset{\_}{F}}{\sqrt{( {\Delta + \overset{\_}{F}} )^{2} - ( {{i\;\omega_{n}} + \overset{\_}{G} + {H\; m_{J}}} )^{2}}}}}} & ({A7})\end{matrix}$with τ being the elastic scattering time and m _(J)′ p_(F)R₃ being theangular momentum cutoff. In the limit H→0, the leading order correctionsto the above equations appear in quadratic order since linear termsvanish due the averaging over m_(J). Indeed, one can show that theself-consistent solution for τ→0 is given by:

$\begin{matrix}{\overset{\_}{G} = {\frac{i}{2\tau}\sin\mspace{11mu} z}} & ({A8}) \\{{\overset{\_}{F} = {\frac{i}{2\tau}\cos\mspace{11mu} z}}{\frac{\omega_{n}}{\Delta} = {{\tan\mspace{11mu} z} - {\kappa\mspace{11mu}\sin\mspace{11mu} z}}}} & ( {{A9},{A10}} )\end{matrix}$where κ=3H²τ

m_(J) ²

/Δ is the characteristic scale for the magnetic field effects in theproblem. Here

$\langle m_{J}^{2} \rangle = {{\frac{1}{m_{J}}{\sum\limits_{{m_{J}} < m_{J}^{2}}\; m_{J}^{2}}} \sim {( {p_{F}R_{3}} )^{2}.}}$Thus, corrections to the pairing gap are governed by the small parameterκ<<1. In terms of the flux quantum, this condition reads

$\frac{\Phi}{\Phi_{0}} ⪡ {R_{3}/{\xi_{d}.}}$Note that disorder suppresses orbital effects of the magnetic field andleads to a weaker dependence of the pairing gap on magnetic field (i.e.,quadratic vs linear). In other words, the disordered superconductor cansustain much higher magnetic fields compared to the clean one, see FIG.20. Finally, the analysis above can be extended to n≠6. After somemanipulations, one finds that:

$\frac{{\Delta(\Phi)} - \Delta_{0}}{\Delta_{0}} \sim {\frac{\xi_{d}^{2}}{R_{3}^{2}}( {n - \frac{\Phi}{\Phi_{0}}} )^{2}}$

This estimate is consistent with the numerical calculations, see FIG.20.

Appendix B: Derivation of the Effective Hamiltonian

In the previous section was derived the Green's function for thedisordered superconducting ring. One can now use these results to studythe proximity effect of the SC ring on the semiconducting core. Considerhere the case when the SC shell is thin d˜l_(e) such that

$\frac{\xi_{d}}{R_{3}} ⪡ 1.$In this case, one can neglect magnetic field dependence of theself-energy for the entire lobe. (Alternatively, when ξ_(d)˜R₃ one canneglect magnetic field effect when

$ {{n - \frac{\Phi}{\Phi_{0}}} ⪡ 1} ).$Thus, one can use zero field Green's functions for the disorderedsuperconductor to investigate the proximity effect which are obtained bysubstituting ω_(n)→{tilde over (ω)}_(n)η(ω_(n)) and Δ₀→{tilde over(Δ)}₀=Δ₀η(ω_(n)) with η(ω_(n))=1+½τ√{square root over (ω_(n) ²+Δ_(n) ²)}in the clean Green's functions.

One can now integrate out the superconducting degrees of freedom andcalculate the effective self-energy due to the tunneling betweensemiconductor and superconductor. Using the gauge convention when Δ₀ isreal, tunneling Hamiltonian between SM and SC is given by:H _(t) =∫drdr′T(r,r′)e ^(inφ/2)Ψ^(†)(r)Ψ(r′)+H.c.  (B1)where r and r′ refer to the SM and SC domains, respectively. T(r, r′) isthe tunneling matrix element between the two subsystems, and Ψ and Ψ^(†)are the fermion annihilation and creation operators in the correspondingsubsystem. One can calculate the SC self-energy due to tunneling tofind:

$\begin{matrix}{{\sum\limits^{({SC})}( {r,\omega_{n}} )} = {{\Gamma(r)}\frac{{i\;\omega_{n}\tau_{0}} - {\Delta_{0}\lbrack {{{\cos( {n\;\varphi} )}\tau_{x}} + {{\sin( {n\;\varphi} )}\tau_{y}}} \rbrack}}{\sqrt{\omega_{n}^{2} + \Delta_{0}^{2}}}}} & ({B2})\end{matrix}$where Γ(r) is a quickly decaying function away from r=R₂ describingtunneling between the two subsystems. Note that the SC self-energy inthis approximation is the same as for a clean superconductor because theratio of {tilde over (ω)}_(n)/{tilde over (Δ)}_(n) is independent of τ.

The Green's function for the semiconductor can be written as:G ⁻¹(ω_(n))=−iω _(n) −H _(SM)−Σ^((SC)(r,ω _(n))  (B3)

In order to calculate quasiparticle energy spectrum one has to find thepoles of above Green's function.

In the hollow cylinder limit, Γ(r=R₂) is a constant and one can find lowenergy spectrum analytically. Indeed, after expanding Equation (B3) insmall ω_(n), the quasiparticle poles are determined by the spectrum ofthe following effective Hamiltonian:

$\begin{matrix}{H_{eff} = {{\frac{H_{SM}}{1 + {\Gamma/\Delta_{0}}} + {\frac{\Gamma}{1 + {\Gamma/\Delta_{0}}}\lbrack {{{\cos( {n\;\varphi} )}\tau_{x}} + {{\sin( {n\;\varphi} )}\tau_{y}}} \rbrack}} = 0}} & ({B4})\end{matrix}$

By comparison with Equation (2), one can establish the correspondencebetween the renormalized and bare parameters of the semiconductor andproximity-induced gap Δ=Δ₀Γ/(Δ₀+Γ).

Appendix C: Effect of Higher m_(j) States on the Gap

As demonstrated above, states with larger m_(j)≠6 have the potential toclose the gap and thus limit the extent of the topological phase. Herewe provide analytical estimates within the hollow cylinder model for theregions in parameter space that become gapless due to higher m_(j)states. For a start, consider the finite m_(j) extension of the BdGHamiltonian (7):

$\begin{matrix}{{{\overset{\sim}{H}}_{m_{j}} = {{\lbrack {\frac{p_{z}^{2}}{2m^{*}} - \mu_{m_{j}}} \rbrack\tau_{z}} + {V_{Z}\sigma_{z}} + A_{m_{j}} + {C_{m_{j}}\sigma_{z}\tau_{z}} + {\alpha\; p_{z}\sigma_{y}\tau_{z}} + {\Delta\;\tau_{x}}}},} & ( {C\; 1} )\end{matrix}$with:

$\begin{matrix}{{\mu_{m_{j}} = {\mu - {\frac{1}{8m^{*}R^{2}}( {{4\; m_{j}^{2}} + 1 + \phi^{2}} )} - \frac{\alpha}{2R_{2}}}}{{V_{Z} = {\phi( {\frac{1}{4m^{*}R_{2}^{2}} + \frac{\alpha}{2R_{2}}} )}},{A_{m_{j}} = {- \frac{\phi\; m_{j}}{4m^{*}R_{2}^{2}}}},{C_{m_{j}} = {- {m_{j}( {\frac{1}{2m^{*}R_{2}^{2}} + \frac{\alpha}{R_{2}}} )}}},}} & {\;( {{C\; 2},{C\; 3},{C\; 4},\;{C\; 5}} )}\end{matrix}$where it is defined that φ=n−Φ(R₂)/Φ₀.

Observe from Eq. (C1) that the pairing gap in the different m_(j)sectors does not necessarily open around zero energy but is shifted bythe τ_(z) independent terms Δ_(mj)+V_(Z)σ_(z) which act as a Paulilimiting field and lead to pairbreaking effects. In other words, thecorresponding m_(J)‥6 particle and hole bands have different Fermimomenta which precludes opening a pairing gap in the spectrum.

One can obtain illustrative analytical estimates for the condition whengapless states are present when considering Hamiltonian (C1) in theabsence of the pairing term Δτ_(x) and the spin orbit termαp_(z)σ_(y)τ_(z). Note that disregarding the spin mixing term slightlyoverestimates the presence of gapless states since the latter allows toopen a gap when particle and hole bands of different σ_(z) componentmix. In fact, the αp_(z)α_(y)τ_(z) is crucial to estimate thetopological gap in the m_(j)=0 sector. For m_(j)≠6, however, themisalignment of the bands at the Fermi level becomes the dominanteffect.

It is possible to estimate the presence of gapless states when both ofthe following conditions are fulfilled for any m_(j)≠6 with spineigenvalue α_(z)=±1.|A _(mj) +V _(Z)σ_(z)>β₁Δ  (C6)−μ_(mj) +V _(z)σ_(z) +A _(mj) +C _(mj)σ_(z)<−β₂Δ,  (C7)where β₁,β₂ are numerical factors in the interval [0,1]. Condition (C6)determines whether particle and hole band crossing is further away thanβ₁Δ from the chemical potential. Assuming a maximally efficient pairingwith corresponding gap ˜Δ can be captured by setting β₁=1. The condition(C7) concerns the position of the bottom of the band of a particularm_(j) and spin state. For Δ=0 the band is at least partially occupiedwhen (C7) holds with β₂=0. In the presence of paring the particle andhole bands repel each other, which shifts the bottom of the particleband to higher energies as compared to the Δ=0 case. This effect can bequalitatively captured by choosing a finite β₂<1.

One can understand the generally finite extent of the gaped regions inthe α-μ plane by observing that condition (C6) is either met forsufficiently large m_(j) or sufficiently large α (when m_(j) is keptconstant). At the same time, large m_(j) states generally violatecondition (C7) since the bottom of the band is shifted up ∝m² _(j) whichneeds to be compensated by sufficiently large μ. It is thereforeexpected to find gapless states for large μ (which enable large m_(j))or very large α which fulfill condition (C6) while still beingcompatible with condition (C7).

These considerations are validated in FIG. 21, which shows the goodagreement between the boundaries predicated by Equations (C6) and (C7)with the numerical simulation of the gap reported in FIG. 16 (using β₁=1and β₂=½). In particular, the figure reveals that the m_(J)=±2 sectorsare those which most limit the extent of the topological phase at largeμ and α. Note that the slope of the blue line in FIG. 21 is independentof β₁ and β₂.

FIG. 21 shows the phase diagram of the hollow cylinder model as afunction of μ and α, as in the left panel of FIG. 16. The contour linesillustrate the analytical estimate for the boundaries of the gappedregion of the phase diagram. The horizontal and sloped dotted lines atthe bottom of the figure are determined by condition (C6) and (C7) form_(j)=±1, while the upper dashed line corresponds to condition (C7) inthe m_(J)=±2 sector.

Appendix D: Details about the Numerical Simulations of Clean Systems

From the numerical perspective, the solution of Equation (B3) for thepoles is not optimal given that one has to solve the non-linear equationfor ω. Therefore, one can employ an alternative approach in which the SMis coupled to an artificial clean superconductor. One can use theparameters for the superconductor and tunneling Hamiltonian such that inthe end Equation (B3) is reproduced after integrating out the SC degreesof freedom.

To obtain the correct self-energy Equation (B2), the thickness of thesimulated clean superconductor needs to be made significantly largerthan the coherence length. This is achieved by using R₃=3 μm for thesimulations shown in FIGS. 17 and 18. All parameters are chosenindependent of r except:

$\begin{matrix}\begin{matrix}{{\Delta(r)} = \{ {\begin{matrix}0 & {r < R_{2}} \\\Delta & {r \geq R_{2}}\end{matrix},} } \\{\alpha = \{ {\begin{matrix}\alpha & {r < R_{2}} \\0 & {r \geq R_{2}}\end{matrix},} } \\{A_{\varphi} = \{ {\begin{matrix}{{\Phi( R_{2} )}{r/( {2\pi\; R_{2}^{2}} )}} & {r < R_{2}} \\{{\Phi( R_{2} )}/( {2\;\pi\; r} )} & {r \geq R_{2}}\end{matrix}.} }\end{matrix} & ({D1})\end{matrix}$

Here, Φ(R₂) corresponds to the flux penetrating the semiconducting core.In accordance to the arguments above, we simulate the superconductorwithout magnetic field. It is possible to solve Equations (6) with thefinite difference method, using a discretization length of 5 nm.

Appendix E: Full Cylinder Semiconductor Model in the Small Radius Limit

In this section there is considered the full cylinder limit discussedpreviously (R₁→0 in FIGS. 1A and 15). Using an effective model it isdemonstrated analytically that the topological phase exists when exactlyone superconducting flux quantum penetrates the core. The results ofthis section are complimentary to the numerical calculations of the maintext above. The effective Hamiltonian for the model is given by:

$\begin{matrix}{{{\overset{\sim}{H}}_{B\; d\; G} = {{( {\frac{p_{z}^{2}}{2\; m^{*}} - {\frac{1}{2\; m^{*}r}\frac{\partial}{\partial r}r\frac{\partial}{\partial r}} - \mu} )\tau_{z}} + {\frac{1}{2\; m^{*}r^{2}}( {m_{J} - {\frac{1}{2}\sigma_{z}} - {\frac{1}{2}n\;\tau_{z}} + {\frac{b}{2}\frac{r^{2}}{R_{2}^{2}}\tau_{z}}} )^{2}\tau_{z}} - {\frac{\alpha}{r}\sigma_{z}{\tau_{z}( {m_{J} - {\frac{1}{2}\sigma_{z}} - {\frac{1}{2}n\;\tau_{z}} + {\frac{b}{2}\frac{r^{2}}{R_{2}^{2}}\tau_{z}}} )}} + {\alpha\; p_{z}\sigma_{y}\tau_{z}} + {{\Delta(r)}\tau_{x}}}},} & ({E1})\end{matrix}$

and, unlike in the hollow cylinder limit, one has to solve the radialpart of Equation (E1). There was introduced the dimensionless variableb=eBR₂ ²=πBR₂ ²/Φ₀. The proximity-induced gap Δ(r) must vanish in themiddle of the core, lim_(r→0) Δ(r)=0. We consider below the case whenΔ(r)=Δr/R₂, although the particular choice for the radial dependence ofΔ(r) is not important for the demonstration of the existence of thetopological phase.

Consider the analysis restricted to the m_(J)=0 sector for n=1 in thelimit

$\frac{1}{\;{m^{*}R_{2}^{2}}} ⪢ {\alpha/{R_{2}.}}$In this case, the problem at hand can be simplified since theHamiltonian (7) becomes separable at α→0 and effect of spin-orbit can beincluded perturbatively. In the limit α→0, the electron spin isconserved and the Bogoliubov transformation diagonalizing Hamiltonian(E1) can written as:

$\begin{matrix}{\gamma_{\lambda,p_{z},\sigma} = {\int_{0}^{R_{2}}{r\; d\;{r\lbrack {{{U_{\lambda,p_{z},\sigma}(r)}{\Psi_{p_{z},\sigma}(r)}} + {{V_{\lambda,p_{z},{- \sigma}}(r)}{\Psi_{p_{z},{- \sigma}}^{\dagger}(r)}}} \rbrack}}}} & ({E2})\end{matrix}$where the transformation coefficients U_(λ,pz,σ)(r) and V_(λ,pz,σ)(r)are given by the solution of Equation (E1). Neglecting the spatialdependence of Δ(r), the functions U_(λ,pz,σ)(r) and V_(λ,pz,σ)(r) can beapproximately written as:Uλ,pz,σ(r)=uλ,σ(pz)fλ,σ(r)  (E3)Vλ,pz,σ(r)=vλ,σ(pz)fλ,σ(r)  (E4)where the single-particle wave functions f_(λ,σ)(r) are defined by thefollowing radial Schrödinger equation:

$\begin{matrix}{{{- \frac{1}{2\; m^{*}}}( {{\frac{1}{r}\frac{\partial}{\partial r}r\frac{\partial}{\partial r}} - \frac{1 + \sigma_{z}}{2\; r^{2}} - {\frac{b^{2}}{4\; R_{2}^{2}}\frac{r^{2}}{R_{2}^{2}}} + {\frac{b}{R_{2}^{2}}\frac{1 + \sigma_{z}}{2}}} ){f_{\lambda,\sigma}(r)}} = {ɛ_{\lambda,\sigma}{f_{\lambda,\sigma}(r)}}} & ({E5})\end{matrix}$

The linear term in b represents a constant energy shift:

$\begin{matrix}{\delta_{\sigma} = \{ \begin{matrix}\frac{b}{2\; m^{*}R_{2}^{2}} & { \sigma = \uparrow ,} \\0 &  \sigma = \downarrow \end{matrix} } & ({E6})\end{matrix}$

After introducing the dimensionless coordinate x=r/R₂ and thedimensionless energies κ_(λ,σ)=2 m*R₂ ²(ε_(λ,σ)+δ_(σ), the aboveequation becomes:

$\begin{matrix}{{( {{{- \frac{1}{x}}\frac{\partial}{\partial x}x\frac{\partial}{\partial x}} + \frac{1 + \sigma_{z}}{2\; x^{2}} + {\frac{b^{2}}{4}x^{2}}} ){f_{\lambda,\sigma}(x)}} = {\kappa_{\lambda,\sigma}{f_{\lambda,\sigma}(x)}}} & ({E7})\end{matrix}$

The normalized eigenstates of this equation, satisfying the boundarycondition f_(λ,σ)(x=1)=0, are:

$\begin{matrix}{{f_{\lambda, \uparrow}(r)} = {C_{\lambda \uparrow}R_{2}^{- 1}x\; e_{1}^{{- x^{2}}/4}{F_{1}( {{1 - \frac{\kappa_{\lambda \uparrow}}{2\; b}},2,\frac{x^{2}}{2}} )}}} & ({E8}) \\{{f_{\lambda, \downarrow}(r)} = {C_{\lambda \downarrow}R_{2}^{- 1}ɛ_{1}^{{- x^{2}}/4}{{F_{1}( {{\frac{1}{2} - \frac{\kappa_{\lambda \downarrow}}{2\; b}},1,\frac{x^{2}}{2}} )}.}}} & ({E9})\end{matrix}$

Here, ₁F₁ is the is the Kummer confluent hypergeometric function and thecoefficients C_(λσ) are determined by the normalization condition:

$\begin{matrix}{{\int_{0}^{R_{2}}{{{f_{\lambda,\sigma}(r)}}^{2}r\; d\; r}} = 1.} & ({E10})\end{matrix}$

The corresponding eigenvalues are:

$\begin{matrix}{ɛ_{\lambda,\sigma} = {\frac{\kappa_{\lambda\sigma}^{2}}{2\; m^{*}R_{2}^{2}} - \delta_{\sigma}}} & ({E11})\end{matrix}$

where κ_(λ,σ) are zeros of the appropriate Kummer confluenthypergeometric function for the two spins. Taking all into account, forb=1 and n=1 the lowest eigenvalues of Equation (E5) are:

$\begin{matrix}\begin{matrix}{{ɛ_{1, \uparrow} \approx \frac{13.77}{2\; m^{*}R_{2}^{2}}},} & {ɛ_{1, \downarrow} \approx \frac{5.84}{2\; m^{*}R_{2}^{2}}} \\{{ɛ_{2, \uparrow} \approx \frac{48.30}{2\; m^{*}R_{2}^{2}}},} & {ɛ_{2, \downarrow} \approx {\frac{30.55}{2\; m^{*}R_{2}^{2}}.}}\end{matrix} & ( {{E12},{E13}} )\end{matrix}$

Note that different values of b will affect the numerical coefficientsreported above.

In the limit

${\frac{1}{2\; m^{*}R_{2}^{2}} ⪢ {\alpha/R_{2}}},\Delta,$one can project the system to the lowest energy manifold (i.e. λ=1) andintegrate over radial coordinate. After some algebra, the effectiveHamiltonian takes the simple form (up to a constant):

${\overset{\sim}{H}}_{B\; d\; G} = {{( {\frac{p_{z}^{2}}{2\; m^{*}} - \overset{\sim}{\mu}} )\tau_{z}} + {{\overset{\sim}{V}}_{Z}\sigma_{z}} + {\overset{\sim}{\alpha}p_{z}\sigma_{y}\tau_{z}} + {\overset{\sim}{\Delta}\tau_{x}}}$where the effective parameters are given by:

$\begin{matrix}{\overset{\sim}{\mu} = {\mu - \frac{ɛ_{1, \uparrow} + ɛ_{1, \downarrow}}{2} - {\frac{\alpha}{2\; R_{2}}( {A_{\uparrow} - \frac{B_{\uparrow} - B_{\downarrow}}{2}} )}}} & ({E14}) \\{{{\overset{\sim}{V}}_{Z} = {\frac{ɛ_{1, \uparrow} - ɛ_{1, \downarrow}}{2} + {\frac{\alpha}{2\; R_{2}}( {A_{\uparrow} - \frac{B_{\uparrow} + B_{\downarrow}}{2}} )}}}{{\overset{\sim}{\alpha} = {\alpha\; C}},{\overset{\sim}{\Delta} = {\Delta\;{D.}}}}} & ( {{E15},{E16},{E17}} )\end{matrix}$with numerical constants A_(σ), B_(σ), C, D given in terms of theoverlap integrals:

$\begin{matrix}{A_{\sigma} = {{\int_{0}^{1}{{{f_{1,\sigma}(x)}}^{2}d\; x}} = \{ {\begin{matrix}{2.056\mspace{14mu}\ldots} &  \sigma = \uparrow  \\{3.521\mspace{14mu}\ldots} &  \sigma = \downarrow \end{matrix},{B_{\sigma} = {{\int_{0}^{1}{x^{2}{{f_{1,\sigma}(x)}}^{2}d\; x}} = \{ {{{\begin{matrix}{0.552\mspace{14mu}\ldots} &  \sigma = \uparrow  \\{0.423\mspace{11mu}\ldots} &  \sigma = \downarrow \end{matrix}\mspace{20mu} C} = {{\int_{0}^{1}{{f_{1, \uparrow}(x)}{f_{1, \downarrow}(x)}d\; x}} = {0.93\mspace{14mu}\ldots}}}\mspace{14mu},\mspace{79mu}{D = {{\int_{0}^{1}{x^{2}{f_{1, \uparrow}(x)}{f_{1, \downarrow}(x)}d\; x}} = {0.465\mspace{14mu}{\ldots\mspace{14mu}.}}}}} }}} }} & ( {{E18},{E19},{E20},{E21}} )\end{matrix}$

One can notice that the Zeeman term remains finite at b=1 (i.e. one fluxquantum) in contrast to the hollow cylinder model. As mentioned in themain text, this is because the semiconducting states are distributedthrough the semiconducting core rather than localized at r=R₂, so thatthe flux cannot perfectly cancel the effect of the winding number.

Thus it has been shown that full cylinder model also maps onto Majoranananowire model (e.g. of Lutchyn et al) and supports topologicalsuperconducting phase. The topological quantum phase transition from thetopologically trivial (i.e. s-wave) to non-trivial (i.e. p-wave) phasesoccurs at:

$\begin{matrix}{{\overset{\sim}{V_{Z}}} = {\sqrt{\overset{\sim}{\mu^{2}} + \overset{\sim}{\Delta^{2}}}.}} & ({E22})\end{matrix}$

Note that this has only considered m_(J)=0 sector. To investigate otherm_(J) sectors and make sure that quasiparticle gap does not close in thetopological phase, this can be done numerically, as shown previously inthe main text.

Appendix F: Topological Phase Transition

FIG. 22 shows probability density for the lowest-energy spin-up (dashed)and spin-down (solid) modes.

FIG. 23 shows (a) DOS in the middle of the wire as a function of fluxfor the same parameters as in FIGS. 18(b), and (b) DOS at the end of thewire as a function of flux for μ=1.5 meV and α=40 meVnm.

In order to understand the topological phase transition within as afunction of magnetic flux within the same lobe, it's useful to study thebulk DOS calculated, for example, in the middle of the wire. Considerthe n=1 lobe in FIG. 23(a) where the topological phase transitionmanifests itself by closing of the bulk. It's also enlightening tocompare the bulk DOS and local DOS at the ends of the wire shown in FIG.18. One may notice the asymmetry with respect to the center of the lobeswhich follows from the different dependence of the semiconducting andsuperconducting states on magnetic field.

This asymmetry depends on parameters and in FIG. 23(b) is shown theboundary DOS for a different set, in which the zero bias peaks extendthroughout the entire n=1 and n=3 lobes. Note, however, that accordingto FIG. 17(a) the system is gapless for this parameters at Φ(R₂)=Φ₀.However, as discussed in relation to FIG. 19A (a), therotational-symmetry-breaking perturbations (e.g. disorder) may lead togap opening for m_(J) Estates and therefore stabilize the topologicalphase.

Appendix G: Details about the Numerical Simulations of DisorderedSystems

For the simulations with disorder in FIGS. 19A-B, the Kwant package hasbeen used for discretizing the Hamiltonian Equation (2) on a 2D squarelattice, using a lattice spacing of 10 nm. The system is assumed to betranslation-invariant along the z direction, with the disorderconfiguration repeating along the z-axis. This trick is required since afull 3D simulation would be computationally too demanding. Toaccommodate to the higher computational cost we use a smaller R₃ of 1.5μm in these simulations. FIG. 24A shows phase diagrams for differentdisorder realizations used to obtain the average in FIG. 19A. FIG. 24Bshows phase diagrams for different disorder realizations used to obtainthe average in FIG. 19B.

CONCLUSION

It will be appreciated that the above embodiments have been described byway of example only.

More generally, according to one definition of the first aspect of thepresent disclosure there may be provided a device comprising one or moresemiconductor-superconductor nanowires, each comprising a length ofsemiconductor material and a coating of superconductor material coatedon the semiconductor material; wherein each of one, some or all of thenanowires is a full-shell nanowire, the superconductor material beingcoated around a full perimeter of the semiconductor material along someor all of the length of the semiconductor material; and wherein thedevice is operable to induce at least one Majorana zero mode, MZM, inone or more active ones of the nanowires including at least one or moreof the full-shell nanowires by application of a magnetic field componentparallel to the active nanowires.

In embodiments, the device may be operable to induce the MZMs at leastby application of the magnetic field component from outside the device.

The device may comprise a substrate and one or more layers formed overthe substrate, wherein the nanowires may be formed in one or more ofsaid layers.

In embodiments, one, some or all of the full-shell nanowires may bevertical relative to the substrate. The device may comprise at least onelayer of filler material disposed between the vertical nanowires tomechanically support the vertical nanowires.

In embodiments one, some or all of the full-shell nanowires may behorizontal in the plane of the substrate.

In embodiments, the device may further comprise one or more layers ofcircuitry formed in one or more of said layers, for connecting thenanowires together into quantum structures, controlling the nanowires orquantum structures, and/or taking measurements from the nanowires orquantum structures.

In embodiments, one or more of the nanowires other than active nanowiresmay be arranged as conductive vias between layers of the circuitry or alayer of the circuitry an exterior of the device.

In embodiments, the one or more layers of circuitry may comprise asemiconducting network that connects the active nanowires, for openingor closing tunnel junctions in the semiconducting network.

In embodiments, the device may comprise one or more qubits, each qubitcomprising a plurality of the active nanowires, wherein one, some or allof the plurality of nanowires in each qubit are full-shell nanowires. Insome such embodiments, each of the qubits may be either: a tetron qubitin which said plurality is four, or hexon qubit in which said pluralityis six.

In embodiments, each qubit may comprises a horizontal superconductingisland formed in a plane paralleled to the substrate, wherein thesuperconducting island is divided into arms each joining the lower endof a respective one of the plurality of vertical nanowires in the qubitto a common point of the superconducting island. In some suchembodiments the arms may take the form of concentric spiral arms.

In embodiments, the superconducting island of each qubit may be formedof the same superconductor material as the coating of the nanowires.

According to another definition of the first aspect there may beprovided a method as defined of operating the disclosed device, themethod comprising: applying the magnetic field component parallel to theactive nanowires in order to induce the at least one MZM in each of theactive nanowires; wherein said inducement comprises a winding of asuperconducting phase of the superconductor material, introduced by amagnetic flux of the magnetic field component through the activenanowires, coupling to the magnetic field component in order to induce atopological phase by means of an orbital effect of the magnetic fieldcomponent. The coupling acts as if a strong Zeeman field was present.

The method may further comprise refrigerating the device to inducesuperconductivity in the superconductor material.

According to one definition of the second aspect there may be provided adevice comprising: a substrate defining a plane; one or more layersformed over the substrate; and one or more semiconductor-superconductornanowires formed in one or more of the layers; wherein each of thenanowires comprises a length of semiconductor material and a coating ofsuperconductor material coated on at least part of the semiconductormaterial; and wherein each of one, some or all of the nanowires isvertical relative to the plane of the substrate.

In embodiments each of one, some or all of the nanowires may be afull-shell nanowire, the superconductor material being coated around afull perimeter of the semiconductor material along some or all of thelength of the semiconductor material.

In embodiments, the device may be operable to induce at least one MZM inone or more active ones of the nanowires by application of a magneticfield component parallel to the active nanowires.

In embodiments, an MZM may be formed at each end of each activenanowire.

In embodiments, the active nanowires may comprise one or more of thefull-shell nanowires.

In embodiments the device may comprise one or more qubits, each qubitcomprising a plurality of the active nanowires.

In embodiments one, some or all of the plurality of nanowires in eachqubit may be full-shell nanowires.

In embodiments one, some or all of the qubits may be MZM-based qubits.In some such embodiments each of one, some or all of the MZM-basedqubits may be either: a tetron qubit in which said plurality is four, orhexon qubit in which said plurality is six.

In embodiments each qubit may comprises a horizontal superconductingisland formed in a plane paralleled to the substrate, wherein thesuperconducting island is divided into arms each joining the lower endof a respective one of the plurality of vertical nanowires in the qubitto a common point of the superconducting island. In some suchembodiments the arms may take the form of concentric spiral arms.

In embodiments the device may further comprise one or more layers ofcircuitry formed in one or more of said layers, for connecting thenanowires together into quantum structures, controlling the nanowires orquantum structures, and/or taking measurements from the nanowires orquantum structures.

In embodiments, one or more of non-active ones the nanowires may bearranged as conductive vias between layers of the circuitry or a layerof the circuitry an exterior of the device.

In embodiments one, some or all of the qubits may be transmon or gatemonbased qubits.

In embodiments, the layers of the wafer may comprise at least one layerof filler material disposed between the vertical nanowires tomechanically support the vertical nanowires.

According to another definition of the second aspect, there may beprovided a method of fabricating a device, the method comprising:providing a substrate; growing vertical lengths of semiconductormaterial perpendicular to the substrate; subsequently coating at least apart of each of at least some of the vertical lengths of semiconductormaterial with a superconductor material, thus forming verticalsemiconductor-superconductor nanowires; and leaving the verticalnanowires in the vertical orientation relative to the substrate uponfinishing the device.

In embodiments the method may comprise, subsequent to said growth,forming a circuit around the vertical nanowires to form one or morequbits, each qubit comprising a plurality of the vertical nanowires.

In embodiments said finishing of the device may comprise any one, moreor all of: forming a gate or contact at the top end of each of some ofor all of the vertical nanowires; depositing a filler material betweenthe vertical nanowires; and/or packaging the device in an integratedcircuit package.

In embodiments said growth may be performed using a vapour-liquid-solid,VLS, growth process.

In embodiments the method may be used to fabricate a device inaccordance with any embodiment of the first or second aspect disclosedherein.

In either the first or second aspect, by way of an exampleimplementation, the semiconductor material may be InAS or InSb. Thesuperconductor material may be Al or Nb.

The superconducting island connecting the nanowires of each qubit may beformed of the same superconductor material as the coating of thenanowires. The filler material is a plastic or a wax. The gates,contacts and/or circuitry between nanowires may be formed from a metalor semiconductor. The at least one MZM induced in each active nanowiremay be a pair of MZMs. In the case of the vertical qubit designs, thelower of the pair may by the operative MZM of the qubit. The device maybe packaged in an integrated circuit package.

In embodiments there may be provided a quantum computer comprising: thedisclosed device of any embodiment of the first or second aspect, and anelectromagnet arranged to apply the magnetic field component parallel tothe active nanowires. The quantum computer may comprise a refrigeratedchamber in which the device is placed in order to inducesuperconductivity in the superconductor.

Other applications or variants of the disclosed techniques or structuresmay become apparent to a person skilled in the art once given thedisclosure herein. The scope of the present disclosure is not limited bythe above described embodiments.

The invention claimed is:
 1. A device comprising one or moresemiconductor-superconductor nanowires, each comprising a length ofsemiconductor material and a coating of superconductor material coatedon the semiconductor material; wherein each of one, some or all of thenanowires is a full-shell nanowire, the superconductor material beingcoated around a full perimeter of the semiconductor material along someor all of the length of the semiconductor material; and wherein thedevice is operable to induce at least one Majorana zero mode, MZM, inone or more active ones of the nanowires including at least one or moreof the full-shell nanowires by application of a magnetic field componentparallel to the active nanowires.
 2. The device of claim 1, wherein thedevice is operable to induce the MZMs at least by application of themagnetic field component from outside the device.
 3. The device of claim1, wherein the device comprises a substrate and one or more layersformed over the substrate, wherein the nanowires are formed in one ormore of said layers.
 4. The device of claim 3, wherein one, some or allof the full-shell nanowires are vertical relative to the substrate. 5.The device of claim 4, further comprising at least one layer of fillermaterial disposed between the vertical nanowires to mechanically supportthe vertical nanowires.
 6. The device of claim 3, wherein one, some orall of the full-shell nanowires are horizontal in the plane of thesubstrate.
 7. The device of claim 3, further comprising one or morelayers of circuitry formed in one or more of said layers, for connectingthe nanowires together into quantum structures, controlling thenanowires or quantum structures, and/or taking measurements from thenanowires or quantum structures.
 8. The device of claim 7, wherein oneor more of the nanowires other than active nanowires are arranged asconductive vias between layers of the circuitry or a layer of thecircuitry an exterior of the device.
 9. The device of claim 7, whereinthe one or more layers of circuitry comprise a semiconducting networkthat connects the active nanowires, for opening or closing tunneljunctions in the semiconducting network.
 10. The device of claim 1,wherein the device comprises one or more qubits, each qubit comprising aplurality of the active nanowires, wherein one, some or all of theplurality of nanowires in each qubit are full-shell nanowires.
 11. Thedevice of claim 10, wherein each of the qubits is either: a tetron qubitin which said plurality is four, or hexon qubit in which said pluralityis six.
 12. The device of claim 4, wherein: the device comprises one ormore qubits, each qubit comprising a plurality of the active nanowires,wherein one, some or all of the plurality of nanowires in each qubit arefull-shell nanowires; and each qubit comprises a horizontalsuperconducting island formed in a plane paralleled to the substrate,wherein the superconducting island is divided into arms each joining thelower end of a respective one of the plurality of vertical nanowires inthe qubit to a common point of the superconducting island.
 13. Thedevice of claim 12, wherein the arms take the form of concentric spiralarms.
 14. The device of claim 12, wherein the superconducting island ofeach qubit is formed of the same superconductor material as the coatingof the nanowires.
 15. The device of claim 1, wherein the device ispackaged in an integrated circuit package.
 16. The device of anypreceding claim, wherein one or more of: the semiconductor material isInAS or InSb, the superconductor material is Al or Nb, and/or the fillermaterial comprises a plastic or wax.
 17. A quantum computer comprising:the device of claim 1; and an electromagnet arranged to apply themagnetic field component parallel to the active nanowires.
 18. Thequantum computer of claim 17, comprising a refrigerated chamber in whichthe device is placed in order to induce superconductivity in thesuperconductor.
 19. A method of operating the device of any of claim 1,the method comprising: applying the magnetic field component parallel tothe active nanowires in order to induce the at least one MZM in each ofthe active nanowires; wherein said inducement comprises a winding of asuperconducting phase of the superconductor material, introduced by amagnetic flux of the magnetic field component through the activenanowires, coupling to the magnetic field component in order to induce atopological phase by means of an orbital effect of the magnetic fieldcomponent.
 20. The method of claim 19, comprising refrigerating thedevice to induce superconductivity in the superconductor material.